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Generic Modeling Framework for Gas Separations Using Multibed Pressure Swing Adsorption Processes Dragan Nikolic,† Apostolos Giovanoglou,‡ Michael C. Georgiadis,§ and Eustathios S. Kikkinides*,† Department of Engineering and Management of Energy Resources, UniVersity of Western Macedonia, SialVera & Bakola Str., 50100 Kozani, Greece, Thermi Business Incubator, Process Systems Enterprise Ltd, 9th Km Thessaloniki-Thermi, 57001 Thessaloniki-Thermi, Greece, Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, London SW7 2AZ, U.K.
This work presents a generic modeling framework for the separation of gas mixtures using multibed pressure swing adsorption (PSA) processes. Salient features of the model include various mass, heat, and momentum transport mechanisms, gas valve models, complex boundary conditions, and realistic operating procedures. All models have been implemented in the gPROMS modeling environment and a formal and user-friendly automatic procedure for generating multibed configurations of arbitrary complexity has been developed. The predictive power of the developed modeling framework for various PSA multibed configurations has been validated against literature and experimental data. The effect of various operating conditions on the product purity and recovery is systematically analyzed. Typical trade offs between capital and operating costs are revealed. 1. Introduction Pressure swing adsorption (PSA) is a gas separation process which has attracted increasing interest because of its low energy requirements as well as low capital investment costs in comparison to the traditional separation processes.1,2 Major industrial applications of PSA include small to intermediate scale oxygen and nitrogen production from air, small to large-scale gas drying, hydrogen recovery from different petrochemical processes (steam-methane reforming off gas or coke oven gas), and trace impurity removal from contaminated gases. From an operational point of view, PSA is an intrinsic dynamic process operating in a cyclic manner with each bed undergoing the same sequence of steps. The typical operating steps include pressurization with feed or pure product, highpressure adsorption, pressure equalization(s), blowdown, and purge. Over the last three decades several PSA studies have appeared in the literature. An overview of single-bed PSA studies is presented in Table 1. Industrial practice indicates that difficult gas separations under high product quality requirements (i.e., purity and/or recovery) rely on complex PSA flowsheets with several interconnected beds and complicated operating procedures. The key literature contributions involving two and multibed configurations under certain assumptions are summarized in Tables 2 and 3. One of the earliest attempts to realistically describe bed interactions is presented by Chou and Huang3 and is based on incorporating a gas valve equation into the PSA model to control flowrate. The results obtained are in good agreement with experimental data. The proposed approach was adopted on the study of the dynamic behavior of two and four bed PSA processes in a subsequent work.4 Warmuzinski and Tanczyk5 developed a generic multibed PSA model. The main features of their approach include sup* To whom correspondence should be addressed. Tel.: +30 2461056650. Fax: +30 24610-56601. E-mail:
[email protected]; kikki@ uowm.gr. † University of Western Macedonia. ‡ Process Systems Enterprise Ltd. § Imperial College London.
port for multicomponent mixtures, linear driving force (LDF) approximation, one or two adsorbent layers and linear pressure changes. The model was used to study the separation of coke oven gas in a five-bed eight-step PSA process. In a subsequent publication,6 the same authors presented an experimental verification of the model using a case study concerning hydrogen recovery from a H2/N2/CH4/CO mixture. In both cases satisfactory agreement between theoretical predictions and experimental data was achieved. Nilchan and Pantelides7 proposed a rigorous mathematical programming approach for the simulation and optimization of PSA. Their work is the first key contribution toward a formal framework for optimizing a PSA process. Cho and co-workers8 developed a mathematical model for the separation of H2 from steam methane reformer off gas. They investigated the effects of carbon-to-zeolite ratio on adsorber dynamics. In a subsequent study,9 they investigated the purification of hydrogen from cracked gas mixture (H2/CH4/CO/CO2) using a double-layered four-bed eight-step PSA configuration. Secchi and co-workers10 studied the modeling and optimization of an industrial PSA unit for hydrogen purification involving six beds and twelve steps. The effects of step durations on process performance were systematically analyzed. Kostroski and Wankat,11 examined the concept of combined cyclic-zone and conventional Skarstrom PSA cycles to increase the product recovery of the latter. Extensive simulations studies were performed using ADSIM, a commercial software product of ASPEN Technology, Inc., and it was showed that the combined cycle can achieve high product recovery while maintaining productivity.11 Sircar and Golden12 presented an overview of industrial PSA processes for the simultaneous production of hydrogen and carbon dioxide, and for the production of ammonia synthesis gas. The selection of adsorbents for hydrogen purification was also investigated. Configurations of four, nine (two series of adsorbents), and ten beds have been analyzed in detail. Waldron and Sircar13 carried out a parametric study of multibed PSA systems for the separation of hydrogen from a H2/CH4 mixture. Three different PSA configurations were employed and the
10.1021/ie0712582 CCC: $40.75 © 2008 American Chemical Society Published on Web 04/02/2008
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 3157 Table 1. Overview of Single-Bed PSA Studies
3 8 27 28 30 18 19 39 44 48 49 55 57 61 63 65 66 a
heat balance
mass balancea
ref
LDF LDF PD LEQ, SD, PD BDPD PD PD 4 different LDF LDF DG LDF DG, LDF -DG, LDF-DGSD, LDF-DGSD LDF LDF LDF Bi-LDF PD
momentum balance
bulk gas
Ergun
bulk gas bulk gas bulk gas bulk gas, solids bulk gas bulk gas
Ergun Ergun
bulk gas bulk gas bulk gas bulk gas, solid
Darcy Ergun Ergun
isotherm, adsorbent
no. steps
no. layers
application
linear, CMS Langmuir, AC + 5A Langmuir, alumina IAS/LRC, AC Langmuir, 5A linear, MS RS-10 linear, MS RS-10 linear, 13X Langmuir/Freundlich, 5A Langmuir, 5A Langmuir, AC + 5A Langmuir, 4A, 5A, CMS
4
4
1 2 1 1 1 1 1 1 1 1 2 1
air separation H2, CO2, CH4, CO air separation H2, CO2, CH4 H2, CH4 air separation, theoretical study air separation, experimental study air separation H2, CO H2, CO2 air separation H2, CO2, CH4, CO air separation
4 4 4 5 4
1 1 1 1 1
CO2 sequestr. CO2 capture H2, CH4 propane, propylene gas drying
4 5 5 4 4 5
Langmuir, 13X dual-site Langmuir, 13X Langmuir, AC Langmuir, 4A linear
LEQ-local equilibrium model; LDF-linear driving force model; SD-solid diffusion model; PD-pore diffusion model; DG-dusty gas model.
Table 2. Overview of Two-Bed PSA Studies ref
mass balance
heat balance
3 7 14 21 29 32 50 64 67
LDF LDF, PD LDF LDF, DG LDF, PD LDF LDF LDF exp
momentum balance Darcy Ergun Darcy
bulk gas bulk gas bulk gas bulk gas bulk gas bulk gas
Ergun Ergun
isotherm, adsorbent
no. beds
no. steps
bed interactions
Langmuir, CMS/5A linear-Langmuir, 5A dual-site Langmuir, 13X Langmuir LRC, AC linear, CMS Langmuir/Freundlich, AC+5A LRC, AC + 5A X zeolite
2 1, 2 1, 2 2 2 2 2 2 2
4 2, 6 3, 6 4 4 6 7 6 4
valve eq valve eq valve eq frozen solid
no. layers
application
1 1 inert + ads. 1 1 1 2 1 1
air separation air separation air separation air separation H2, CH4 air separation H2, CH4, CO, N2, CO2 air separation air separation
Table 3. Overview of Multibed PSA Studies ref
mass balance
heat balance
momentum balance
isotherm, adsorbent
no. beds
Langmuir, 5A LRC, AC + 5A LRC, AC/5A Langmuir, AC + 5A
4 5 2, 4 4
6 8 5, 7 8
Langmuir, alumina + AC + zeolite AC, zeolite, silica gel
6
12
9, 10
9, 11, 6 + 7
Langmuir, AC dual-site Langmuir, APHP + 5A ext. Langmuir, 13X Langmuir, 13X
4, 5 5
7, 9, 12 11
2, 3 4, 5
4, 6 4, 5, 5
4 5 6 9
LEQ LDF LDF LDF
bulk gas bulk gas bulk gas
Ergun
10
LDF
bulk gas
Darcy
12 13 15
LDF LDF
bulk gas bulk gas
52 68
LEQ LDF
bulk gas bulk gas
Darcy Ergun
no. steps
effect of numerous process variables on the separation quality was evaluated. Biegler and co-workers14-16 developed a robust simulation and optimization framework for multibed PSA processes. They investigated the effect of different operating parameters on process performance using a five-bed eleven-step PSA configuration for the separation of hydrogen from H2/CH4/N2/CO/ CO2 mixture. Bed interactions have been described using gas valve equation. To simulate the behavior of a five-bed PSA configuration two different approaches were employed: the “unibed” and the “multibed”.15 The unibed approach assumes that all beds undergo identical steps so only one bed is needed to simulate the multibed cycle. Information about the effluent streams are stored in data buffers and linear interpolation is used to obtain information between two time points. The multibed
bed interactions valve eq linear change
no. layers
application
1 1 or 2 1 2
air separation H2, CH4, CO, N2 H2, CH4, N2 H2, CO2, CH4, CO
inert + 2 ads
H2, CH4, CO
valve eq
1 2
overview of commercial PS processes H2, CH4 N2, CH4, CO2, CO, H2,
linear change
1 1
CO2 from flue gases CO2, N2, H2O
only one bed is simulated; information stored in data buffer information stored in data buffer
approach considers a multibed process as a sequence of repetitive stages within the cycle. In most of the previous studies either complex mathematical models and simple bed arrangements, or simple mathematical models and complex bed arrangements have been presented. Mass balance is usually represented by the LDF approximation and heat balance by the typical bulk gas equation. Detailed mass and heat transfer mechanisms at the particle level are often neglected. Transport properties such as mass and heat transfer coefficients, gas diffusivities (molecular, Knudsen, surface), and mass and heat axial dispersion coefficients are often assigned to constant values. Furthermore, most of the previous contributions focus on a relatively small number of units with simple bed interactions (frozen solid model, the linear pressure change or the information about streams are stored in data buffers for
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all feasible PSA cycle step configurations and interbed connectivities. The gPROMS17 modeling and optimization environment has been served as the development platform. It must be emphasized that there have been several commercial software products in the market for some time focused on the simulation of PSA operations, such as ADSIM by Aspen Technology Inc. This product is well-known and documented for its power to simulate PSA processes of various degrees of complexity.11 However, it is beyond the objective of this work to be compared with ADSIM or any similar commercial product simply because the latter is a standard simulation package with existing PSA libraries, while in our work we only use the mathematical equation-based environment of gPROMS to build our own PSA models and then to create an integrated framework to perform systematic simulation and optimization studies of multibed PSA systems. To this end, general model equations and boundary conditions are developed, both single- and multibed configurations are studied and a systematic approach to automatically generate feasible operating procedures in complex PSA flowsheets is proposed. 2. Mathematical Modeling
Figure 1. Single-bed PSA arrangement with all possible interconnections.
later use). Moreover, simulations of multibed PSA processes are usually carried out by simulating only one bed with the exception of the works of Sircar and Golden12 and Jiang et al.15 In the present work, a generic modeling framework for multibed PSA flowsheets is been developed. The framework is general enough to support an arbitrary number of beds, a customized complexity of the adsorbent bed model, one or more adsorbent layers, automatic generation of operating procedures,
Figure 2. Four-bed PSA flowsheet.
The proposed framework consists of several mathematical models, operating procedures, and auxiliary applications presented in Appendix A. 2.1. Adsorbent Bed Models. The mathematical modeling of a PSA column has to take into account the simultaneous mass, heat and momentum balances at both bed and adsorbent particle level, adsorption isotherm, transport and thermophysical properties of the fluids and boundary conditions for each operating step. The architecture of the developed adsorbent bed model sets a foundation for the problem solution. Internal model organization defines the general equations for the mass, heat and momentum balances, adsorption isotherm, and transport and
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 3159 Table 5. Geometrical and Transport Characteristics of PSA Process
Figure 3. Algorithm for the generation of feasible operating procedures. Table 4. Comparison of the Experimental and the Simulation Results recovery, % no.
exp19
model19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 42 43
6.44 8.16 10.65 12.39 7.07 9.41 11.67 12.99 25.84 21.36 22.74 22.18 9.39 7.40 5.10 9.19 7.07 6.43 5.41 11.51 9.72 6.74 3.37 16.63 10.76 8.57 8.26 7.35 11.95 13.30 41.56 22.46 12.54 13.83 23.05 22.40 22.37
6.38 8.64 10.92 12.15 8.25 9.85 11.72 12.13 17.02 16.09 15.50 22.74 8.35 6.32 5.00 8.59 7.62 6.99 6.13 12.44 9.67 6.59 4.08 14.33 11.55 8.47 7.21 8.56 14.27 16.07 37.64 16.07 8.00 13.09 16.73 16.46 16.15
purity, % this work
exp19
model19
this work
6.96 8.50 9.94 11.30 8.52 9.95 10.72 11.41 21.23 21.21 21.20 21.08 8.45 7.85 7.38 9.82 8.56 7.42 6.12 11.28 9.43 7.26 5.46 10.86 9.62 8.81 8.44 11.35 11.17 11.20 26.38 15.11 8.99 9.86 14.25 20.51 20.06
99.75 99.55 99.00 98.35 99.60 98.85 98.30 97.80 90.80 90.70 91.40 91.20 98.25 98.90 99.00 98.45 98.50 98.70 98.80 97.95 98.50 98.85 99.00 95.85 96.80 97.30 97.65 98.75 97.70 97.00 82.30 93.70 98.20 93.80 93.70 89.00 82.30
100.00 99.89 99.09 98.59 99.88 99.41 98.77 98.49 95.39 96.23 96.18 96.02 99.70 99.82 99.88 99.63 99.76 99.83 99.93 98.71 99.42 99.79 99.96 97.47 98.72 99.15 99.10 99.87 97.92 96.57 82.27 94.97 100.00 94.45 97.58 94.40 89.72
99.31 98.92 98.55 98.15 98.95 98.54 98.33 98.11 94.47 94.75 94.78 94.77 99.74 99.91 99.97 99.60 99.79 99.89 99.96 98.25 99.40 99.91 99.99 98.69 99.57 99.84 99.91 98.22 98.31 98.24 89.23 96.70 99.06 96.06 97.75 94.96 90.12
thermophysical properties. Phenomena that occur within the particles could be described by using many different transport mechanisms. However, only the mass and heat transfer rates through a particle surface in the bulk flow mass and heat balance equations have to be calculated. Hence, the same adsorbent bed
parameter
value
temperature of the feed temperature of the purge gas temperature of the wall feed pressure purge pressure gas valve constant particle density particle radius bed void fraction bed inner diameter bed length molar fraction of H2 in feed molar fraction of CO in feed molar fraction of CH4 in feed molar fraction of CO2 in feed overall mass transfer coefficient of H2 overall mass transfer coefficient of CO overall mass transfer coefficient of CH4 overall mass transfer coefficient of CO2 heat capacity of the particles heat transfer coefficient (wall) heat axial dispersion coefficient mass axial dispersion coefficient
297 297 297 26.3 × 105 1 × 105 1 × 10-5 850 3 × 10-3 0.36 0.1 1.0 0.755 0.040 0.035 0.170 1.0 0.3 0.4 0.1 878.64 138.07 1 × 10-3 1 × 10-3
units K K K Pa Pa kg/m3 m m m
1/s 1/s 1/s 1/s J/(kg‚K) W/(m2‚K) m2/s m2/s
Table 6. Langmuir Parameters and Heat of Adsorption for Activated Carbon comp
a1 × 103 (mol/g)
a2 (K)
b1 × 109 (Pa-1)
b2 (K)
-∆Hads (J/mol)
H2 CO CH4 CO2
4.32 0.92 -1.70 -14.20
0.00 0.52 1.98 6.63
5.040 5.896 19.952 24.775
850.5 1730.9 1446.7 1496.6
7861.73 16677.42 18752.69 25572.61
Table 7. Four-Bed Six-Step PSA Configuration (Pressurization with Feed, One Pressure Equalization Step, Purge from Tank) Bed 1 Bed 2 Bed 3 Bed 4
CoCP Purge Blow Ads
Ads EQR1 Purge EQD1
Ads CoCP Purge Blow
EQD1 Ads EQR1 Purge
Blow Ads CoCP Purge
Purge EQD1 Ads EQR1
Purge Blow Ads CoCP
EQR1 Purge EQD1 Ads
Table 8. Eight-Bed Eight-Step PSA Configuration (Pressurization with Feed, Two Pressure Equalization Steps, Purge from Tank) Bed 1 Bed 2 Bed 3 Bed 4 Bed 5 Bed 6 Bed 7 Bed 8
CoCP EQR1 EQR2 Purge Blow EQD2 EQD1 Ads
Ads CoCP EQR1 EQR2 Purge Blow EQD2 EQD1
EQD1 Ads CoCP EQR1 EQR2 Purge Blow EQD2
EQD2 EQD1 Ads CoCP EQR1 EQR2 Purge Blow
Blow EQD2 EQD1 Ads CoCP EQR1 EQR2 Purge
Purge Blow EQD2 EQD1 Ads CoCP EQR1 EQR2
EQR2 Purge Blow EQD2 EQD1 Ads CoCP EQR1
EQR1 EQR2 Purge Blow EQD2 EQD1 Ads CoCP
model could be used for implementing any transport mechanism as long as it follows the defined interface that calculates transfers through the particle surface. This way, different models describing mass and heat transfer within the particle can be plugged into the adsorbent column model. Systems where the mass transfer in the particles is fast enough can be described by the local equilibrium model, thus avoiding the generation of thousands of unnecessary equations. On the other hand, more complex systems may demand mathematical models taking into account detailed heat and mass transfer mechanisms at the adsorbent particle level. They may even employ equations for the phenomena occurring at the molecular level without any changes in the macro level model. This is also the case for other variables such as pressure drop, transport, and thermophysical properties of gases that can be calculated by using any available equation. The result of such architecture is the fully customizable adsorbent bed model that can be adapted for the underlying application to the desired level of complexity.
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Table 9. Twelve-Bed Ten-Step PSA Configuration (Pressurization with Feed, Three Pressure Equalization Steps, Purge from Tank) Bed 1 Bed 2 Bed 3 Bed 4 Bed 5 Bed 6 Bed 7 Bed 8 Bed 9 Bed 10 Bed 11 Bed 12
CoCP EQR1 EQR2 EQR3 Purge Purge Blow EQD3 EQD2 EQD1 Ads Ads
Ads CoCP EQR1 EQR2 EQR3 Purge Purge Blow EQD3 EQD2 EQD1 Ads
Ads Ads CoCP EQR1 EQR2 EQR3 Purge Purge Blow EQD3 EQD2 EQD1
EQD1 Ads Ads CoCP EQR1 EQR2 EQR3 Purge Purge Blow EQD3 EQD2
EQD2 EQD1 Ads Ads CoCP EQR1 EQR2 EQR3 Purge Purge Blow EQD3
Table 10. Input Parameters of Run 1
EQD3 EQD2 EQD1 Ads Ads CoCP EQR1 EQR2 EQR3 Purge Purge Blow
Blow EQD3 EQD2 EQD1 Ads Ads CoCP EQR1 EQR2 EQR3 Purge Purge
Purge Blow EQD3 EQD2 EQD1 Ads Ads CoCP EQR1 EQR2 EQR3 Purge
Purge Purge Blow EQD3 EQD2 EQD1 Ads Ads CoCP EQR1 EQR2 EQR3
EQR3 Purge Purge Blow EQD3 EQD2 EQD1 Ads Ads CoCP EQR1 EQR2
1
SPPressCoC SPAdsIn SPAdsOut SPBlow SPPurgeIn SPPurgeOut SPPEQ1 SPPEQ2 SPPEQ3 feed flowrate, m3/s purge flowrate, m3/s τPress, s τAds, s τBlow, s τPurge, s τPEQ1, s τPEQ2, s τPEQ3, s τcycle, s
EQR1 EQR2 EQR3 Purge Purge Blow EQD3 EQD2 EQD1 Ads Ads CoCP
Table 12. Runtime Parameters for Run II
number of beds parameter
EQR2 EQR3 Purge Purge Blow EQD3 EQD2 EQD1 Ads Ads CoCP EQR1
4
number of beds
8
12
0.1000 0.3600 0.0035 0.8000 0.2250 0.0300
0.060 0.3600 0.0015 0.5000 0.2250 0.0300 0.0040
0.030 0.3600 0.0015 0.2500 0.2250 0.030 0.0025 0.0050
3.60 × 10-6
3.60 × 10-6
3.60 × 10-6
0.0500 0.3600 0.0015 0.4000 0.2250 0.0300 0.0030 0.0040 0.0080 3.60 × 10-6
2.25 × 10-6
2.25 × 10-6
2.25 × 10-6
2.25 × 10-6
10 20 10 20
10 20 10 20 10
20 20 20 20 20 20
60
80
160
10 20 10 20 10 10 10 120
Table 11. Simulation Results of Run I
parameter
1
4
8
12
SPPressCoC 0.0300 0.0450 0.0500 0.0500 SPAdsIn 0.1200 0.1300 0.3300 0.3600 SPAdsOut 0.0004 0.00035 0.0014 0.0015 SPBlow 0.2000 0.2200 0.3000 0.4000 SPPurgeIn 0.5625 0.1500 0.3000 0.2250 SPPurgeOut 0.0075 0.0200 0.0400 0.0300 SPPEQ1 0.0025 0.0020 0.0300 SPPEQ2 0.0040 0.0040 SPPEQ3 0.0080 feed flowrate, 3.30 × 10-6 3.60 × 10-6 1.20 × 10-6 1.30 × 10-6 m3/s purge flowrate, 1.15 × 10-6 3.00 × 10-6 2.25 × 10-6 0.75 × 10-6 m3/s τPress, s 20 15 15 10 τAds, s 40 30 15 20 τBlow, s 20 15 15 10 τPurge, s 40 30 15 20 τPEQ1, s 15 15 10 τPEQ2, s 15 10 τPEQ3, s 10 τcycle, s 120 120 120 120
Table 13. Simulation Results for Run II
no. beds
H2 purity (%)
H2 recovery (%)
power (W)
productivity (mol/kg‚s)
no. beds
H2 purity (%)
H2 recovery (%)
power (W)
productivity (mol/kg‚s)
1 4 8 12
96.87 99.50 99.58 99.36
34.40 58.06 68.87 72.50
5290.2 2794.9 1198.7 1393.6
9.18 × 10-2 4.85 × 10-2 2.08 × 10-2 2.42 × 10-2
1 4 8 12
98.11 99.29 99.79 99.36
20.21 38.40 65.25 72.50
1442.0 1406.9 1409.2 1393.6
2.50 × 10-2 2.44 × 10-2 2.45 × 10-2 2.42 × 10-2
The following general assumptions have been adopted in this work: (i) the flow pattern in the bed is described by the axially dispersed plug flow (no variations in radial direction across the adsorber); (ii) the adsorbent is represented by uniform microporous spheres and at the particle level, only changes in the radial direction occur. The following generic features have been implemented. Mass transfer in particles is described by four different diffusion mechanisms: (i) local equilibrium (LEQ), (ii) linear driving force (LDF), (iii) surface diffusion (SD), (iv) pore diffusion
(PD). Three different thermal operating modes of the adsorber have been implemented: isothermal, nonisothermal, and adiabatic. Momentum balance is described by Blake-Kozeny’s or Ergun’s equation. Multicomponent adsorption equilibrium is described by Henry’s law, extended Langmuir type isotherm or by using the ideal adsorbed solution theory (IAS). Thermophysical properties are calculated using the ideal gas equation or an equation of state. Transport properties are assigned constant values or predicted using appropriate correlations. In this work, the Langmuir isotherm has been applied as a pure com-
Figure 4. Simulation results of run I.
Figure 5. Simulation results of run II.
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Figure 6. Simulation results of run III. Table 14. Runtime Parameters for Run III number of beds parameter
1
4
8
12
SPPressCC 0.0030 0.0030 0.0500 0.0040 SPAdsIn 0.4000 0.3333 0.3450 0.5000 SPAdsOut 0.00088 0.00085 0.0015 0.0014 SPBlow 0.2500 0.4000 0.3000 0.4000 SPPurgeIn 0.0900 0.1333 0.3000 0.2000 SPPurgeOut 0.0140 0.0200 0.0400 0.0300 SPPEQ1 0.0050 0.0020 0.0030 SPPEQ2 0.0040 0.0040 SPPEQ3 0.0080 -6 -6 -6 feed flowrate, 3.33 × 10 3.45 × 10 5.00 × 10-6 4.00 × 10 3 m /s purge flowrate, 1.33 × 10-6 3.00 × 10-6 2.00 × 10-6 0.90 × 10-6 m3/s τPress, s 20 15 15 10 τAds, s 40 30 15 20 τBlow, s 20 15 15 10 τPurge, s 40 30 15 20 τPEQ1, s 15 15 10 τPEQ2, s 15 10 τPEQ3, s 10 τcycle, s 120 120 120 120
Table 15. Simulation Results for Run III no. beds
H2 purity (%)
H2 recovery (%)
power (W)
productivity (mol/kg‚s)
1 4 8 12
99.87 99.97 99.97 99.83
1.41 28.47 50.58 65.16
1887.8 1180.8 1193.8 1190.9
2.03 × 10-2 2.05 × 10-2 2.07 × 10-2 2.07 × 10-2
ponent adsorption isotherm to calculate the spreading pressure in the IAS theory. However, any isotherm equation including an expression for spreading pressure, which is explicit in the pressure can be used. Finally, proper boundary conditions for the various operating steps have been developed in accordance with previous studies on multibed configurations. The overall modeling framework is summarized in Appendix A. 2.2. Gas Valve Model. The gas valve model describes a oneway valve. The purpose of the one-way valve is to force the flow only to desired directions and to avoid any unwanted flows. Gas valve equations are also included in Appendix A. 2.3. Multibed PSA Model. The single-bed PSA model provides the basis for the automatic generation of the flowsheet via a network-superstructure of single adsorbent beds. The main building block of the multibed PSA model is illustrated in Figure 1. The central part of the building block is the adsorbent bed model. Feed, purge gas, and strong adsorptive component streams are connected to the corresponding bed ends via gas valves. This is also the case for light and waste product sinks. The bed is properly connected to all other beds in the system at both ends via gas valves. Such a configuration makes the
flowsheet sufficiently general to support all feasible bed interconnections. This way all possible operating steps in every known PSA process can be supported. The main building block can be replicated accordingly through an input parameter representing the number of beds in the flowsheet. A typical fourbed PSA flowsheet is shown in Figure 2. 2.4. Operating Procedures. Procedures for controlling the operation of multibed PSA processes are highly complex due to the large number of interactions. Hence, an auxiliary program for automatic generation of operating procedures has been developed. This program generates operating procedures for the whole network of beds according to the given number of beds and sequence of operating steps in one bed. Operating procedures govern the network by opening or closing the appropriate valves at the desired level and changing the state of each bed. This auxiliary development allows the automatic generation of a gPROMS source code for a given number of beds and sequence of operating steps in one bed. It also generates gPROMS tasks ensuring feasible connectivities between the units according to the given sequence. A simplified algorithm illustrating the generation of operating procedures is presented in Figure 3. The program is not limited only to PSA configurations where beds undergo the same sequence of steps, but it can also handle more complex configurations such as those concerning the production of two valuable components.12 2.5. Numerical Solution. The modeling equations are solved using the gPROMS mathematical framework. More particularly the spatial domains are discretized using several options from finite difference and finite element schemes. Then the resulting system of nonlinear differential-algebraic equations is solved using stiff integrator algorithms based on a sophisticated variable-step variable-order numerical scheme, which can efficiently handle model discontinuities and complex operating procedures. In the present studies the axial domain is discretized using orthogonal collocation on finite elements of third order with 20 elements, while the radial domain in the adsorbent spherical particles is discretized using the same scheme with 5-10 elements depending on the case under consideration. Other discretization algorithms (e.g., central finite difference schemes) have also been tested without showing any significant advantage. It must be noted that in general, the solution of adsorptionbased model equations may cause several complications particularly when diffusive or dispersive terms have minimal effect or are absent. This is due to the fact that the resulting partial differential equations become purely hyperbolic causing nonphysical oscillations and numerical instability in the standard discretization schemes. In such cases the easiest remedy is to employ standard first order upwind finite difference schemes,11 an option that is available in the gPROMS environment. However this will reduce the accuracy of the discretization since the accuracy of first order differencing schemes scales linearly with the length-step, while the inevitable introduction of numerical dispersion caused by the use of these schemes will distort the physical picture of the problem. In general there exist superior discretization methods particularly suited for such cases (e.g., adaptive multiresolution approach20) which overcome instability problems or the divergence of the numerical method in the computed solution in the presence of steep, fast moving fronts. These methods are capable of locally refining the grid in the regions where the solution exhibits sharp features thus allowing nearly constant discretization error throughout the computational domain. This approach has been successfully applied in the simulation and optimization of cyclic adsorption processes.21,22 Nevertheless, in all case studies considered in
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Table A1. Basic Adsorbent Bed Model feature
general equation
bulk flow mass balance bed generation term in bulk flow mass balance (transfer through a particle surface)
(
∀ z ∈ (0, L), i ) 1, ..., Ncomp
Ni ) f (T, P, Ci, adsorbent type, ...), ∂(bedFcpuT) ∂z
bulk flow heat balance
)
∂(uCi) ∂Ci ∂Ci ∂ D , + bed + (1 - bed)Ni ) bed ∂z ∂t ∂z z,i ∂z ∀ z ∈ (0, L), i ) 1, ..., Ncomp
+
3kh,wall ∂(bedFcpT) (T - Twall) ) + (1 - bed)qg + ∂t Rbed ∂ ∂T bed λz , ∀ z ∈ (0, L) ∂z ∂z
(A1)
(A2) (A3)
( )
generation term in bulk flow heat balance (transfer through a particle surface) momentum balance overall mass transfer coefficient overall heat transfer coefficient mass axial dispersion coefficient heat axial dispersion coefficient
∂P ) f (u, bed, Rp, ...), ∀ z ∈ (0, L) ∂z kf,iRp Sh ) ) f (Sc, Re), ∀ z ∈ [0, L], i ) 1, ..., Ncomp Dm,i -
Nu )
khRp ) f (Pr, Re), λ
Dz,ibed Dm,i λz λ
) f (Sc, Re),
∀ z ∈ [0, L]
∀ z ∈ [0, L], i ) 1, ..., Ncomp
∀ z ∈ [0, L], i ) 1, ..., Ncomp
P ) f (T, Ci, ..., CNcomp),
thermophysical properties
F, cp, λ, µ ) f (T, P, Ci, ..., CNcomp), ∂Ci , ∂z ∂T Fcpu(T - Tin) ) λz , ∂z u(Ci - Cin i ) ) Dz,i
∂u ) 0, ∂z boundary conditions for stream outlet from the bed closed bed end
∂Ci ∂z
(A4) (A5) (A6) (A7) (A8) (A9)
∀ z ∈ [0, L]
) f (Pr, Re),
equation of state
boundary conditions for stream inlet into the bed
∀ z ∈ (0, L), i ) 1, ..., Ncomp
qg ) f (T, P, Ci, adsorbent type, ...),
∀ z ∈ [0, L], i ) 1, ..., Ncomp
z ) 0 or z ) L, i ) 1, ..., Ncomp z ) 0 or z ) L
z ) 0 and z ) L
(A10) (A11) (A12) (A13) (A14) (A15)
) 0,
∂T ) 0, ∂z
the present work, the contribution of diffusive/dispersive terms is significant ensuring a parabolic nature in the model partial differential equations and thus all numerical simulations did not reveal any nonphysical oscillations. Therefore the orthogonal collocation on finite elements is considered in the present study to be an adequate discretization scheme in terms of accuracy and stability. 3. Model Validation The single-bed pore diffusion model is validated against the results of Shin and Knaebel.18,19 The first part of their work presents a pore diffusion model for a single column.18 They developed a dimensionless representation of the overall and component mass balance for the bulk gas and the particles, and applied Danckwert’s boundary conditions. In this work the effective diffusivity, axial dispersion coefficient, and mass transfer coefficient, are taken from the work of Shin and Knaebel. Phase equilibrium is given as a linear function of gasphase concentration (Henry’s law). The design characteristics of the adsorbent and packed bed are also adopted from the work of Knaebel.18,19 The axial domain is discretized using orthogonal collocation on finite elements of third order with 20 elements. The radial domain within the particles is discretized using the same method of third order with five elements. Shin and Knaebel used six collocation points for discretization of the axial domain
z ) 0 or z ) L, i ) 1, ..., Ncomp z ) 0 or z ) L
(A16)
and eight collocation points for the radial domain. The bed is considered to be clean initially. The target is to separate nitrogen from an air stream using molecular sieve RS-10. Here, oxygen is the preferably adsorbed component due to higher diffusivity. The PSA configuration consists of one bed and four operating steps (pressurization with feed, high-pressure adsorption, countercurrent blowdown, and countercurrent purge). The effect of the following operational variables on the N2 purity and recovery is systematically analyzed: duration of adsorption step for a constant feed velocity, feed velocity (for a fixed duration of adsorption step), duration of adsorption step for a fixed amount of feed, duration of blowdown step, purge gas velocity for a fixed duration of purge step, duration of purge step for a constant purge gas velocity, duration of purge step for a fixed amount of purge gas, duration of pressurization step, feed/purge step pressure ratio, and column geometry (length/ diameter ratio). The pore diffusion model presented in section 2 and Appendix A has been used for comparison purposes. Gas valve constants and corresponding stem positions have been properly selected to produce the same flow rates as in the work of Shin and Knaebel.19 Since during the simulations several parameters have been changed, corresponding modifications in stem positions of the gas valves have been made. The stem positions of the gas valves for the base case are 0.15 for feed inlet, 0.002 for
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 3163 Table A2. Mass Transport Mechanisms model
equation mass transfer between a bulk gas and particles is assumed instantaneous and concentration of the gas within the particles is equal to the concentration in the bulk flow: ∀ r ∈ [0, Rp],
) Ci,
Cpi
(A17)
∀ z ∈ [0, L], i ) 1, ..., Ncomp
the gas phase is in equilibrium with the solid phase: LEQ
Qi ) Q/i ,
∀ r ∈ [0, Rp],
(A18)
∀ z ∈ [0, L], i ) 1, ..., Ncomp
the generation term is sum of accumulation in gas phase in particles and amount adsorbed: ∂Q/i ∂Ci N i ) Fp + p , ∂t ∂t
(A19)
∀ z ∈ (0, L), i ) 1, ..., Ncomp
mass transfer between a bulk gas and particles is assumed instantaneous: Cpi ) Ci,
LDF
r ) Rp,
(A20)
∀ z ∈ [0, L], i ) 1, ..., Ncomp
concentration profile within the particle is assumed parabolic: ∂Q h i 15De,i / (Qi - Q h i), r ∈ [0, Rp], ∀ z ∈ [0, L], i ) 1, ..., Ncomp ) ∂t R2
(A21)
the generation term is sum of accumulation in gas phase in particles and amount adsorbed: ∂Q hi ∂Ci N i ) Fp + p , ∀ z ∈ (0, L), i ) 1, ..., Ncomp ∂t ∂t
(A22)
p
the gas phase within particles is neglected: ∀ r ∈ [0, Rp],
Cpi ) 0,
all adsorption happens at the particle surface; the adsorbed phase is then transported toward the particle center by surface diffusion (Fick’s first law of diffusion): ∂Qi ∂Qi 1 ∂ Rp2Ds,i , ∀ r ∈ (0, Rp), ∀ z ∈ [0, L], i ) 1, ..., Ncomp ) 2 ∂t ∂r R ∂r
(A24)
mass transfer between bulk gas and particles is carried out through the gas film around particles: 3kf Ni ) (Ci - Cpi |r)Rp), ∀ z ∈ (0, L), i ) 1, ..., Ncomp Rp
(A25)
p
SD
(A23)
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(
)
boundary conditions: kf,i(Ci - Cpi |r)Rp) ) Ds,i ∂Qi ∂r
) 0,
r ) 0,
|
∂Qi ∂r
,
r)Rp
r ) Rp, i ) 1, Ncomp
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(A26) (A27)
within the pores, gas is transported by combined molecular and Knudsen diffusion: p
(
)
∂Cpi ∂Q/i ∂Cpi 1 ∂ + (1 - p)Fp ) p 2 Rp2De,i , ∂t ∂t ∂r R ∂r p
∀ r ∈ (0, Rp),
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(A28)
the gas phase in micro-pores is in equilibrium with the adsorbed phase: Qi ) Q/i , PD
∀ r ∈ [0, Rp],
(A29)
∀ z ∈ [0, L], i ) 1, ..., Ncomp
mass transfer between bulk gas and particles is carried out through the gas film around particles: 3kf Ni ) (Ci - Cpi |r)Rp), ∀ z ∈ (0, L), i ) 1, ..., Ncomp Rp boundary conditions: kf,i(Ci - Cpi |r)Rp) ) De,i ∂Cpi ∂r
) 0,
r ) 0,
∀ z ∈ [0, L], i ) 1, ..., Ncomp
feed outlet, 0.7 for blowdown, and 0.3 for purge. All the other cases have been modified according to the base case. This was necessary since the flow rate through the gas valve depends linearly on the valve pressure difference for a given stem position (e.g., in order to increase the flowrate through the gas valve the stem position should be linearly increased). Table 4 presents a comparison of experimental and simulation results of this work and the work of Shin and Knaebel.19 The modeling approach presented in this work predicts satisfactorily the behavior of the process and the overall average absolute deviation from experimental data is limited to 1.97% in purity and 2.23% in recovery, while the deviations between Shin and
|
∂Qi ∂r
r)Rp
,
r ) Rp, i ) 1, Ncomp
(A30)
(A31) (A32)
Knaebel’s19 theoretical and experimental results were 1.81% and 2.52% respectively. These small differences can be attributed to the use of a gas valve equation employed to calculate the flowrate and pressure at the end of the column, as opposed to linear pressure histories during pressure changing steps used in the original work of Shin and Knaebel.19 It should be emphasized that the gas valve equation results in exponential pressure histories during the pressure changing steps. Moreover, different number of discretization points and different package for thermo-physical property calculations have been used compared to the work of Shin and Knaebel.19
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Table A3. Thermal Operating Modes mode
equation ∂T ) 0, ∀ z ∈ (0, L) ∂z temperature within the particles are constant and equal to the temperature of bulk flow (no heat transfer between bulk flow and particles occurs): Tp ) T, ∀ r ∈ [0, Rp], ∀ z ∈ [0, L]
isothermal
(A33)
(A34)
bulk flow heat balance: ∂(bedFcpuT) ∂z LEQ model
+
∂(bedFcpT) 3kh,wall ∂ ∂T (T - Twall) ) bed λz , + qg + ∂t Rbed ∂z ∂z
( )
∀ z ∈ (0, L)
heat transfer between bulk gas and particles is assumed instantaneous and the temperature of particles is equal to the temperature of the bulk flow: ∀ r ∈ [0, Rp],
Tp ) T,
(A35)
(A36)
∀ z ∈ [0, L]
the generation term qg is given by the sum of accumulations in gas and solid phases within the particles and heat of adsorption/desorption: qg ) LDF model
∂[(Fpcpp + pFpgcp,pg)T]
∂Q/i , ∂t
Ncomp
+ Fp
∂t
∑
(∆Hads,i)
i
(A37) ∀ z ∈ (0, L)
heat transfer between bulk gas and particles is assumed to be instantaneous and the temperature of particles is equal to the temperature of the bulk flow: ∀ r ∈ [0, Rp],
Tp ) T,
(A38)
∀ z ∈ [0, L]
the generation term qg is given by the sum of accumulations in gas and solid phases within the particles and heat of adsorption/desorption: qg ) nonisothermal
SD model
∂[(Fpcpp + pFpgcp,pg)T]
∂Q hi , ∂t
Ncomp
+ Fp
∂t
∑ (∆H i
ads,i)
(A39) ∀ z ∈ (0, L)
heat transfer between bulk gas and particles is assumed to be instantaneous and the temperature of particles is equal to the temperature of the bulk flow: ∀ r ∈ [0, Rp],
Tp ) T,
the generation term qg is given by the sum of accumulation in solid phase within the particles and heat of adsorption/desorption: Ncomp ∂Qi|Rp ∂(FpcppT) qg ) + Fp , ∀ z ∈ (0, L) (∆Hads,i) ∂t ∂t i
(A41)
heat transfer is carried out through the gas film around particles: 3kh qg ) (T - Tp|Rp), ∀ z ∈ (0, L) Rp
(A42)
∑
PD model
(A40)
∀ z ∈ [0, L]
particle heat balance: ∂[(Fpcpp + pFpgcp,pg)Tp] ∂t
+ qpg )
(
)
∂Tp 1 ∂ Rp2λp , 2 ∂r ∂r Rp
∀ r ∈ (0, Rp),
∀ z ∈ (0, L)
(A43)
the generation term qpg is given by the heat of adsorption/desorption: ∂Q/i , ∂t
Ncomp
qpg ) Fp
∑ (∆H
ads,i)
i
( )|
kh(T - Tp|r)Rp) ) λp boundary conditions:
adiabatic
∂Tp ) 0, ∂r
∂Tp ∂r
∀ r ∈ (0, Rp) ,
r)Rp
r ) Rp
r)0
(A44)
(A45) (A46)
the same as in the nonisothermal mode, except the following term is neglected: 3kh,wall Rred
(T - Twall)
4. Case Studies The developed modeling framework has been applied in a PSA process concerning the separation of hydrogen from steammethane reforming off gas (SMR) using activated carbon as an adsorbent. In this case, the nonisothermal linear driving force model has been used for simulation purposes. The geometrical data of a column, adsorbent and adsorption isotherm parameters for activated carbon have been adopted from the work of Park
et al.8 and are given in Tables 5 and 6. The effective diffusivity, axial dispersion coefficients, and heat transfer coefficient are assumed constant in accordance with the work of Park et al.8 Two different cycle configurations have been used. The sequence of the steps differs only in the first step. In the first configuration (configuration A), pressurization has been carried out by using the feed stream whereas in the second (configuration B) by using the light product stream (pure H2). Flowsheets
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 3165 Table A4. Momentum Balance and Equilibrium Isotherms feature
equation Blake-Kozeny equation: 2 ∂P 180µu (1 - bed) ) , ∂z (2R )2 3 p
momentum balance
(A47)
∀ z ∈ (0, L)
bed
Ergun’s equation: -
1 - bed Fu (1 - bed)2 µu|u| ∂P + 1.75 , ) 150 ∂z 3 d2 3 dp bed
p
(A48)
∀ z ∈ (0, L)
bed
linear (Henry type): Q/i ) HiCpi ,
(A49)
∀ z ∈ [0, L], i ) 1, ..., Ncomp, Hi ) f (T, P)
extended Langmuir: biPi Q/i ) Qm,i ,
∑b P
1+
Qm,i ) ai,1 +
ai,2 , T
bi ) bi,1 exp
() bi,2 T
,
∀ z ∈ [0, L], i, j ) 1, ..., Ncomp
(A50)
j j
j
ideal adsorbed solution theory (IAS):2,69
∫
π/i ) equilibrium isotherms
Pi0
/ Qpure,i
P
0
/ π/i ) πi+1 ,
Langmuir
8 π/i ) Qm,i log(1 + biP0i ), dP 9 isotherm
∀ z ∈ [0, L], i ) 1, ..., Ncomp
∀ z ∈ [0, L], i ) 1, ..., Ncomp - 1 biP0i
/ Qpure,i ) Qm,i , 1 + biP0i
Qm,i ) ai,1 +
ai,2 , T
(A51a)
(A51b)
bi ) bi,1 exp
( )
bi,2 , T
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(A51c)
x/i P0i ) xiP,
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(A51d)
∑
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(A51e)
x/i
1 Qtotal
) 1,
)
∑Q i
Q/i ) x/i Qtotal,
x/i
/ pure,i
,
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(A51f)
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(A51g)
Table A5. Calculation of Transport Properties variable
equation correlation:24
overall mass transfer coefficient
Wakao Dm,i kf,i ) (2.0 + 1.1Sc0.33Re0.6), Rp
overall heat transfer coefficient
Wakao correlation:25 λ kh ) (2.0 + 1.1Pr0.33Re0.6), Rp
mass axial dispersion coefficient
heat axial dispersion coefficient
Wakao correlation:24 Dm,i Dz,i ) (20 + 0.5ScRe), bed Wakao correlation:25
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(A53)
∀ z ∈ [0, L]
∀ z ∈ [0, L], i ) 1, ..., Ncomp
(A54) (A55)
∀ z ∈ [0, L]
λz ) λ(7 + 0.5PrRe),
(A52)
Chapman-Enskog equation:2 Dm,i ) 1.8583 × 10 molecular diffusivity
Fuller
x x
1 T3 MWi
-3
(A56) ,
∀ z ∈ [0, L],
∀ r ∈ [0, Rp], i ) 1, 2
,
∀ z ∈ [0, L],
∀ r ∈ [0, Rp], i ) 1, 2
Pσ122Ω12
correlation:2
-7 1.75
Dm,i ) 1.013 × 10 T
1 MWi
PX12
(A57)
2
Kauzmann2 correlation: Knudsen diffusivity
Dk,i ) 9.7 × 103Rpore
effective diffusivity
De,i )
p Dm,iDk,i , τp Dm,i + Dk,i
x
of one, four, eight, and twelve beds have been simulated. These configurations differ only in the number of pressure equalization
T , MWi
∀ z ∈ [0, L],
∀ z ∈ [0, L],
∀ r ∈ [0, Rp], i ) 1, ..., Ncomp
∀ r ∈ [0, Rp], i ) 1, ..., Ncomp
(A58)
(A59)
steps introduced. Thus, the one-bed configuration contains no pressure equalization steps, the four-bed involves one, the eight-
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Table A6. Gas Valve Equation variable
molar flowrate
equation
If Pout > PcritPin:
otherwise:
critical pressure power
x∑
1-
F ) CvSPPin
|
Pout Pin
2
|,
i ) 1, ..., Ncomp
x|∑ |
i ) 1, ..., Ncomp
xiMWiT
1 - (Pcrit)2
F ) CvSPPin
Pcrit )
( )
,
xiMWiT
(1 +2 κ)
power )
(A60)
(A61)
κ/1-κ
( ) (( ) Pfeed κ RTfeed κ-1 Plow
bed configuration two, and the twelve-bed flowsheet involves three pressure equalization steps. The following sequence of steps has been used: pressurization, adsorption, pressure equalization (depressurization to other beds), blowdown, purge, and pressure equalization (repressurization from other beds). The sequence of steps for the one-bed configuration is Ads, Blow, Purge, CoCP, and an overview of operating steps employed in the other configurations is presented in Tables 7-9, where CoCP stands for cocurrent pressurization, Ads stands for adsorption, EQD1, EQD2, and EQD3 are the pressure equalization steps (depressurization to the other bed), Blow represents countercurrent blowdown, Purge is the countercounter purge step, and EQR3, EQR2, and EQR1 are the pressure equalization steps (repressurization from the other bed). Configurations using pressurization by the light product (H2) are the same apart from the first step. All configurations have been generated using the auxiliary program described in section 2. The bed is initially assumed clean. For simulation purposes, the axial domain is discretized using orthogonal collocation on finite elements of third order with 20 elements. Three different sets of simulations have been carried out. (A) Simulation Run I. In this case configuration A has been employed (pressurization with feed). The effect of number of beds and cycle time (due to introduction of pressure equalization steps) on the separation quality has been investigated. The following operating conditions have been selected: constant duration of adsorption and purge steps and constant feed and purge gas flowrates. The input parameters of the simulation are given in Table 10, and the simulation results are summarized in Table 11 and Figure 4. The results clearly illustrate that, as the number of beds increases, an improved product purity (∼3%) and product recovery (∼38%) are achieved. The improved purity can be attributed to the fact that flowsheets with lower number of beds process more feed per cycle (feed flowrate is constant but more feed is needed to repressurize beds from the lower pressure). The noticeable increase in product recovery is the direct result of the pressure equalization steps. On the other hand, power requirements and adsorbent productivity decrease due to the lower amount of feed processed per unit time (the power and productivity of the eight-bed configuration are lower than the power and productivity of the twelve-bed configuration due to higher cycle time). The purity of the twelve-bed configuration is lower than the purity of the four and eight-bed ones. This interesting trend can be attributed to the fact that during the third pressure equalization a small breakthrough takes place thus contaminating the pressurized bed. (B) Simulation Run II. In simulation run II, configuration A has been employed (pressurization with feed). The effect of number of beds for constant power requirements and constant
κ-1/κ
-1
)
Nfeed τcycle
(A62) (A63)
adsorbent productivity on the separation quality has been analyzed. The following operating conditions have been selected: constant adsorbent productivity, constant cycle time, and constant amount of feed processed per cycle. The input parameters are given in Table 12 and the simulation results are summarized in Table 13 and Figure 5. The results show that, increasing the number of beds, a slight increase in the product purity (∼1%) and a significant increase in product recovery (∼52%) are achieved. In this run the amount of feed processed per cycle is constant and improve in the purity cannot be attributed to the PSA design characteristics and operating procedure. Due to the same reasons described in run I, the purity of the twelve-bed configuration is lower than the purity of the eight-bed one. The power requirements and adsorbent productivity remain constant due to the constant amount of feed processed per cycle. (C) Simulation Run III. In this case, configuration B (pressurization with light product) is used. The effect of number of beds for constant power requirements and constant adsorbent productivity on separation quality has been investigated. The following operating conditions have been selected: constant adsorbent productivity, constant duration of adsorption and purge steps, constant feed and purge gas flowrates, constant cycle time, and constant amount of feed processed per cycle. The input parameters are given in Table 14, and the simulation results are presented in Figure 6 and Table 15. The results illustrate that the number of beds does not affect the product purity, productivity and power requirements (since the amount of feed per cycle is constant). However, a huge improvement in the product recovery (∼64%) is achieved due to three pressure equalization steps. Similar to the results in runs I and II, the purity of the twelve-bed configuration is lower than the purity of the eight-bed one. The above analysis reveals typical trade offs between capital and operating costs and separation quality. Thus, by increasing the number of beds, a higher product purity/recovery is achieved while higher capital costs are required (due to a larger number of beds). On the other hand, energy demands are lower due to energy conservation imposed by the existence of pressure equalization steps. 5. Conclusions A generic modeling framework for the separation of gas mixtures using multibed PSA flowsheets is presented in this work. The core of the framework represents a detailed adsorbent bed model relying on a coupled set of mixed algebraic and partial differential equations for mass, heat and momentum balance at both bulk gas and particle level, equilibrium isotherm equations and boundary conditions according to the operating
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 3167
steps. The adsorbent bed model provides the basis for building a PSA flowsheet with all feasible inter-bed connectivities. Operating procedures are automatically generated thus facilitating the development of complex PSA flowsheet for an arbitrary number of beds. The modeling framework provides a comprehensive qualitative and quantitative insight into the key phenomena taking place in the process. All models have been implemented in the gPROMS modeling environment and successfully validated against experimental and simulation data available from the literature. An application study concerning the separation of hydrogen from the steam methane reformer off gas illustrates the predictive power of the developed framework. Simulation results reveal typical trade offs between operating and capital costs. Current work concentrates on the development of a mathematical programming framework for the optimization of PSA flowsheets. Recent advances in mixed-integer dynamic optimization techniques will be employed for the solution of the underlying problems. Future work will also consider the detailed modeling and optimization of hybrid PSA-membrane processes. Acknowledgment Financial support from PRISM EC-funded RTN (Contract number MRTN-CT-2004-512233) is gratefully acknowledged. The authors thank Professor Costas Pantelides from Process Systems Enterprise Ltd for his valuable comments and suggestions. Appendix A Tables A1-A6 contain the equations used in this analysis. Nomenclature a1 ) Langmuir isotherm parameter, mol/kg a2 ) Langmuir isotherm parameter, K b1 ) Langmuir isotherm parameter, 1/Pa b2 ) Langmuir isotherm parameter, K b ) Langmuir isotherm parameter, m3/mol C ) molar concentrations of gas phase in bulk gas, mol/m3 Cin ) molar concentrations of gas phase at the inlet of bed, mol/m3 p C ) molar concentrations of gas phase in particles, mol/m3 cp ) heat capacity of bulk gas, J/(kg‚K) cp,pg ) heat capacity of gas within pellets, J/(kg‚K) cpp ) heat capacity of the particles, J/(kg‚K) Cv ) valve constant De ) effective diffusivity coefficient, m2/s Dk ) Knudsen diffusion coefficient, m2/s Dm ) molecular diffusion coefficient, m2/s dp ) particle diameter, m Ds ) surface diffusion coefficient, m2/s Dz ) axial dispersion coefficient, m2/s F ) molar flowrate, mol/s ∆Hads ) heat of adsorption, J/mol H ) Henry’s parameter, m3/kg kf ) mass transfer coefficient, m2/s kh ) heat transfer coefficient, J/(m2‚K‚s) kh,wall ) heat transfer coefficient for the column wall, J/(m2‚K‚ s) L ) bed length, m MW ) molecular weight, kg/mol Ncomp ) number of components, Nfeed ) number of moles in feed stream, mol Ni ) molar flux through the particle surface, mol/(m2‚s)
Nu ) Nusselt dimensionless number, Pr ) Prandtl dimensionless number, P0 ) pressure that gives the same spreading pressure in the multicomponent equilibrium, Pa Pin ) pressure at the inlet of the gas valve, Pa Pout ) pressure at the outlet of the gas valve, Pa Q ) adsorbed amount, mol/kg Q* ) adsorbed amount in equilibrium state with gas phase (in the mixture), mol/kg qg ) heat generation term, heat flux through particle surface, J/(m2‚s) Qm ) Langmuir isotherm parameter, mol/kg Q/pure ) adsorbed amount in equilibrium state with gas phase (pure component), mol/kg Qtotal ) total adsorbed amount, mol/kg r ) radial discretization domain, m Rbed ) bed radius, m Re ) Reynolds dimensionless number, Rp ) pellet radius, m Rpore ) radius of the pore, m Sc ) Schmidt dimensionless number, Sh ) Sherwood dimensionless number, SP ) stem position of a gas valve SPAdsIn ) stem position of a gas valve during adsorption (inlet valve), SPAdsOut ) stem position of a gas valve during adsorption (outlet valve), SPBlow ) stem position of a gas valve during blowdown, SPPEQ1 ) stem position of a gas valve during pressure equalization, SPPEQ2 ) stem position of a gas valve during pressure equalization, SPPEQ3 ) stem position of a gas valve during pressure equalization, SPPressCoC ) stem position of a gas valve during cocurrent pressurization, SPPressCC ) stem position of a gas valve during countercurrent pressurization, SPPurgeIn ) stem position of a gas valve during purge (inlet valve), SPPurgeOut ) stem position of a gas valve during purge (outlet valve), T ) temperature of bulk gas, K Tin ) temperature of the fluid at the inlet of the bed, K Tp ) temperature of particles, K Twall ) temperature of the wall, K u ) interstitial velocity, m/s x ) molar fraction in gas phase, x* ) molar fraction in adsorbed phase, Z ) compressibility factor, z ) axial discretization domain, m Greek Letters bed ) porosity of the bed, p ) porosity of the pellet, µ ) viscosity of bulk gas, Pa s λ ) thermal conductivity of bulk gas, J/(m‚K) λz ) heat axial dispersion coefficient, J/(m‚K) λp ) thermal conductivity of the particles, J/(m‚K) π* ) reduced spreading pressure, F ) density of bulk gas, kg/m3 Fpg ) density of gas within pellets, kg/m3 Fp ) density of the particles, kg/m3 κ ) heat capacity ratio, -
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ReceiVed for reView September 18, 2007 ReVised manuscript receiVed February 22, 2008 Accepted February 26, 2008 IE0712582
