Optimization of Vacuum Pressure Swing Adsorption Processes To

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Optimization of Vacuum Pressure Swing Adsorption Processes to Sequester Carbon Dioxide from Coalbed Methane Daeho Ko Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b01288 • Publication Date (Web): 27 Jul 2016 Downloaded from http://pubs.acs.org on July 28, 2016

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Industrial & Engineering Chemistry Research

Optimization of Vacuum Pressure Swing Adsorption Processes to Sequester Carbon Dioxide from Coalbed Methane

Daeho Ko*

Global Engineering Division of GS Engineering & Construction Gran Seoul, 33, Jong-ro, Jongno-gu, Seoul 03159, Korea

* To whom correspondence should be addressed. Tel: +82 2 2154 6171, Email: [email protected] or [email protected]

1

ACS Paragon Plus Environment


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Abstract This paper introduces an efficient scale up design technology based on mathematical modeling and optimization approach. The target process is a vacuum pressure swing adsorption process which is preferred in purifying unconventional gas such as coalbed methane or landfill gas by sequestering carbon dioxide. The optimization results are validated very well from a pilot scale (10Nm3/h) to a large commercial scale (5000Nm3/h). The simulation and optimization model for the scale up design is established in gPROMS custom modeling tool. Judging from the reasonably very small differences between the optimization results and the vendor suggested data, it can be concluded that the scale up design is very successful not by depending on the experimental and real plant experiences but by using mathematical optimization modeling approach.

Keywords: simulation, optimization, vacuum pressure swing adsorption, scale-up design, coalbed methane

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1. Introduction Pressure swing adsorption (PSA) processes have been very attractive in industries1 for a wide range of applications over the past three decades especially in the hydrogen purification, air separation, and gas separation/purification processes. PSA processes can be considered as an eco-friendly technology because it does not need the chemicals such as amine solvents of absorption processes but employs regenerable solid adsorbents. Because the adsorbents are regenerated by lowering the operating pressure instead of heating, the energy intensity of PSA technology is low. In addition, the PSA units can be easily downsized to be mounted as modules on a skid which is appropriate for the transportation and exploitation of small gas reservoirs, for example, unconventional gas reservoirs such as coalbed methane (CBM), which has recently been accepted as an important economic energy resource.2 It has been generally claimed that in spite of the growing practical applications of adsorption technology to gas separation an experimental effort is still required in the commercial design and optimization of cyclic adsorption processes (CAPs) such as a PSA and a thermal swing adsorption (TSA)3 due to the inherent complex characteristics and big computational load for simulations. Vetukuri et al. (2010)4 reported the reasons as follows: 1) CAPs are complex systems including indigenous nonlinear partial differential and algebraic equations (PDAEs) with different initial and boundary conditions at each step, 2) multiadsorbent layers, nonisothermal effects, and stringent product specifications may lead to numerical solver failures resulting from the nonlinearities and ill-conditioned matrices, and 3) the overall optimization work is dominated by cyclic steady state (CSS) condition that the variable profiles such as concentration, temperature, and adsorption amount at the beginning of the cycle are identical to those at the end of the cycle. In addition, Ko (2016)5 pointed out that the conventional gas velocity calculation method without tuning based on experimental 3



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experiences makes the PSA simulation model be inaccurate and difficult to be converged in simulations. Thus, optimization of CAPs has still been regarded as a computationally very challenging task. Smith and Westerberg (1990) developed mixed-integer nonlinear programming (MINLP) model to determine the optimal schedule of a PSA process based on a given set of operating steps and constraints – e.g., bed interconnections, continuous operation of compressor, and continuous production.6 They (1991) considered the cyclic operating schedule of a PSA system and presented an MINLP model for optimization with time averaged mass and energy balances in the design of a PSA process for the minimum annualized cost.7 Nilchan and Pantelides (1998) introduced an optimization framework of PSA processes by proposing complete discretization (CD) of the PDAEs and efficiently solving a nonlinear programming (NLP) problem.8 Ding et al. (2002) proposed a direct optimization algorithm with the direct determination of cyclic steady states (CSSs) of PSA and TSA.9 Cruz et al. (2003, 2005) developed an optimization procedure for small and large scale CAP processes,10 and proposed an innovative CAP optimization approach using high-resolution schemes and a grid-adaptation strategy.11 Jiang et al. (2003) employed a direct determination approach by using a Newtonian-based method to accelerate the convergence on CSS for the PSA optimization.12 They (2004) developed a robust simulation and optimization framework for multibed PSA processes and observed the effect of different operating parameters on process performance using a 5-bed 11-step PSA process for the separation of hydrogen from a H2/CH4/N2/CO/CO2 mixture.13 Ko et al. (2003)14 developed a PSA optimization approach which can be easily implemented in gPROMS modeling system using single discretization (SD) instead of complete discretization (CD)8 and showed the approach makes it possible to optimize efficiently the highly nonlinear PDAE-based PSA model at CSS. The same authors 4


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(2005)15 upgraded the CSS determination method from the former research14 in a manner similar to that by Jiang et al. (2003).12 This updated method (2005)15 showed the optimization results with more accurate CSS, better convergence, and faster computation than the former method (2003).14 Agarwal et al. (2009)16 developed a reduced order model (ROM) based on proper orthogonal decomposition (POD) approximating a dynamic PDE based model to a DAE system of significantly lower order, resulting in making optimization problems be computationally efficient. The same authors (2010)17 suggested a systematic optimizationbased formulation to synthesize PSA cycles, that is, a superstructure able to predict a number of different PSA operating steps, for the postcombustion CO2 capture using PSA processes. Nikolic´ et al. (2009)19 introduced an optimization framework for complex PSA processes including multibed configurations and multilayered adsorbents in order to optimize the number of beds, PSA cycle configuration, and various operating and design parameters. Khajuria and Pistikopoulos (2013)18 presented a simultaneous approach of incorporated PSA design, operational, and control problems under process uncertainty such as the effect of time variant and invariant disturbances. In addition to the above optimization tasks, multiobjective optimization studies of adsorption processes have been performed. Ko and Moon (2002)20 modified the summation of a weighted objective function (SWOF) method and achieve the multiobjective optimization of TSA and rapid pressure swing adsorption (RPSA), which is the first application of multiobjective optimization to highly nonlinear CAPs. Sankararao and Gupta (2007)21 developed the multiobjective optimization technique, called the modified MOSA-aJG, that is, multiobjective simulated annealing with a jumping gene algorithm for stochastic search of the decision variables, and carried out multiobjective optimization studies in the operation of air separation PSA units. Fiandaca and Fraga (2009)22 presented a preliminary investigation of 5



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the performance of a custom multi-objective genetic algorithm (MOGA) for the optimization of a fast cycle PSA operation. Haghpanah et al. (2013)23 achieved full multiobjective purityrecovery and energy-productivity Pareto fronts of the analyzed vacuum swing adsorption (VSA) cycles by using genetic algorithm (GA)-based multiobjective optimization algorithm. Though numerous studies have been done on the optimization of cyclic adsorption processes (CAP), to the best of author’s knowledge, there have been no works on the application of an optimization method to the scale up design up to large commercial size CAP plants without experimental efforts. In this study, the optimizations are performed to find and analyze several optimal design conditions of 10Nm3/h scale vacuum pressure swing adsorption (VPSA) processes according to the different constraints and specifications. Then, this work firstly shows the results of an excellent and accurate scale up design technology of VPSA processes, which are more efficient adsorption technology in recovering methane from CBM than PSA processes25, 26, 27, by adopting the state of the art Ko’s mathematical simulation model (2016)5 as well as the novel optimization methodology (2005).15 The results from the current optimization models based on the simulation approach5 are verified from the scale of 10Nm3/h (feed flow) to that of 5000Nm3/h (feed flow). This technology was applied to design a real pilot plant (10Nm3/h) which was constructed in a coalbed methane (CBM) test site.

2. FORMULATION OF TARGET PROCESS 2.1. Target Process. The present study is an optimization work extended from the former simulation research (2016)5 which introduced the firstly developed simulation approach able 6


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to predict the exact interstitial gas velocity within adsorption beds of CAP processes. The accurate gas velocity calculation of CAP processes makes it possible to perform the scale up design without tuning or real plant expertise as explained in Ko’s paper (2016).5 That is to say, the former research (2016)5 reported the newly developed robust and accurate simulation model of VPSA processes and validated the developed simulation model by comparing with laboratory scale experimental data (feed flow: 0.44~0.68Nm3/h) and pilot plant design data (feed flow: 100Nm3/h) provided by PSA design company, Gens Engineering Co. Ltd. in Korea. Based on the model (2016),5 the optimizations for the scale up designs of VPSA processes up to a large commercial size (~5000Nm3/h in feed flow) are performed to attain the basic design conditions in this study. The adopted adsorbent, the operating condition (the operating step and pressures), the adsorption bed configuration, and the feed compositions are same as Ko’s paper (2016).5 In other words, 1) the adsorbent is SHIRASAGI MSC (molecular sieving carbon) 3K-172, made by Japan EnviroChemicals, Ltd. in Japan, 2) The assumed CBM feed composition is 89.5% methane, 8.18% carbon dioxide, and 2.32% nitrogen, and 3) the four-bed twelve-operating step VPSA process is employed as illustrated in Table 15 and Figure 1. The operating sequence is explained in Supporting Information, Figures S1.1 – S1.12, and the adsorption isotherm of this adsorbent is shown in Figure S2.5 2.2. Mathematical Formulation for Simulation. The mathematical model equations are summarized in Table 2. The detailed simulation model is described and evaluated in the former paper (Ko, 2016).5 The key point of the model formulation is in the mole balance (eq 11) which can calculate the accurate interstitial gas velocity profile more efficiently and robustly, leading to an excellent scale up design performance using the optimization method of Ko et al. (2005)15 as well as the Ko’s simulation approach (2016).5 The model assumptions are as follows: 7



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(1) The Redlich-Kwong (RK) equation of state (EOS) is used to describe the gas phase behaviors instead of the ideal gas law assumption.5 (2) Radial variations of temperature, pressure, and concentration are neglected. (3) Competitive adsorption behaviors are described by the Langmuir equation for a mixture of gases. (4) The adsorption rate is approximated by the LDF expression. (5) Physical properties of the bed are independent of the temperature. (6) The gas velocity is calculated by using molar flow rate variable and the molar flow rate is obtained from the novel mole balance equation.5 (7) Pressure profiles within adsorption beds are calculated by Ergun equation.5 (8) The time varying pressure profiles at the boundary of the bed during the pressure changing steps such as pressure equalization steps and pressurization steps can be expressed by the exponential function of time.28 (9) The adsorption beds are in nonisothermal and adiabatic conditions. The accuracy of RK EOS is slightly improved from that of ideal gas law as shown in Supporting Information, Table S1, and the assumed pressure profiles (assumption 8) are verified by comparing the pressure profiles of real pilot plant operation with those of simulation as shown in Figure S3. Because the differences of predicted performances in the adiabatic condition and those in nonadiabatic condition are very small as reported in Supporting Information, Table S3, the adiabatic condition can be assumed in this study.

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To perform the simulation and optimization efficiently, the uni-bed modeling technique is also used in this optimization study. Uni-bed modeling in this study is done by defining the following boundary conditions of mole fractions (eqs 12-14)5 at the interacting steps instead of using data storage buffer13, 29, 30 or virtual-bed concept19.

[y

]

06 = yiStep , ave

[y

]

04 = yiStep , ave

Step 09 i z=L

Step10 i z=L

[y

]

[

]

(12)

[

]

(13)

Steps11−12 i z=L

[

z=L

z=L

02−13 = yiSteps , ave

]

(14)

z=L

As a result, the above concept shows the same effect as the uni-bed approaches of Kumar (1994)29 and Jiang et al.(2004)13, single-interacting-bed (SIB) method of Patel (2014)30, and modified uni-bed approach, called virtual-bed approach of Nikolic´ et al. (2009)19, because the average mole fraction information from the step of the bed supplying the gas – i.e., step 4, step 6, and steps 2-3 – is enough to calculate the mole fractions, mole numbers of each component within the bed receiving the gas at the steps – i.e., step 10, step 9, and steps 11-12 – during the pressure equalization and repressurization steps. The gas contact time, tcont, within the bed at adsorption step is assumed as Lbed uNormal, feed = Lbed Avoid V&Normal, feed for the convenient calculations in simulation and optimization instead of using real contact time ( Lbed u Normal , ave ) where uNormal,ave is average velocity value at normal condition as to both the adsorption time and spatial domain.

3. CSS DETERMINATION FOR OPTIMIZATION MODEL

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This work uses the updated tailored single discretization (uTSD)15 to efficiently carry out the optimization of the VPSA process for the scale up design as well as to satisfy the CSS condition. The CSS condition means that the spatial profiles of mole fraction, adsorption amount, and temperature at the beginning of the cycle are same as those at the end of the cycle. Thus the CSS is defined by eq (15) which can only be expressed by using complete discretization (CD) which makes the model size bigger and be more difficult to converge in the simulation and optimization calculation than single discretization (SD). The advantages of the tailored single discretization (TSD) over CD method were explained by Ko et al. (2003).14

φ ( z )t = 0 = φ ( z )t =t

(15)

cycle

where, 0 < z < L if φ = yi or T

0 ≤ z ≤ L if φ = qi In the uTSD method15 employed in this study, the following constraint (eq 16) is used instead of eq (15).

− ε ≤ φ ( z )t =0 − φ (z )t =t cycle ≤ ε

(16)

Here, ε is a nonnegative variable included in the optimization objective function to be minimized. z in φ ( z )t =0 is discretized to zl, l = 1,…, ND, where ND denotes the number of discretization in finite difference method as to the bed axial domain. The term, φ ( z )t =0 , can be expressed as yCH4,l,t=0, yCO2,l,t=0, yN2,l,t=0, qCH4,l,t=0, qCO2,l,t=0, qN2,l,t=0, and Tl,t=0, which are optimization decision variables to determine the CSS. The CSS condition can be effectively satisfied through the optimization with proper starting points of φ ( z )t =0 for the optimizations 10


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because there are no nonlinear terms in eq (16). In the current optimization models, the pressure and temperature variables as optimization decision variables are normalized to improve the optimization performance. Thus the normalized variable values must be within 0 and 1 as defined in eqs (17)-(22). The values of the normalized decision variables are found as a result of the optimization as well as the corresponding un-normalized variables. With the present normalization technique, which is the similar to the variable scaling method for optimizations31, the computation speed becomes faster, and success rate of optimization convergence is nicely improved. Steps ( 01− 03) Steps ( 01− 03), LB PAD − PAD Steps ( 01− 03) PÂD = Steps Steps ( 01− 03), LB PAD ( 01− 03),UB − PAD

where

(17)

Steps( 01− 03), LB Steps( 01− 03),UB PAD = 5 × 105 Pa, PAD = 15 ×105 Pa

Step ( 04 ) Step ( 04 ). LB PEQ − PEQ 1 1 Step ( 04 ) PÊQ = 1 Step ( 04 ).UB Step ( 04 ). LB PEQ1 − PEQ1

(18)

Step( 04).LB Step( 04).UB = 3 × 105 Pa, PEQ = 11×105 Pa where PEQ 1 1

Step ( 06 ) Step ( 06 ). LB PEQ − PEQ 2 2 Step ( 06 ) PÊQ = 2 Step ( 06 ).UB Step ( 06 ). LB PEQ 2 − PEQ 2

where

(19)

Step( 06).LB Step( 06).UB PEQ = 1.1× 105 Pa, PEQ = 7 × 105 Pa 2 2

Step ( 07 ) Step ( 07 ). LB PBD − PBD Step ( 07 ) PˆBD = Step Step ( 07 ). LB PBD ( 07 ).UB − PBD

where

(20)

Step( 07).LB Step( 07).UB PBD = 0.999 ×105 Pa, PBD = 1.001×105 Pa

11



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Step ( 08 ) Step ( 08 ). LB PPG − PPG Step ( 08 ) PˆPG = Step Step ( 08 ). LB PPG ( 08 ).UB − PPG

where

(21)

Step ( 08 ). LB Step ( 08 ).UB PPG = 0.05 × 105 Pa, PPG = 0.5 × 105 Pa

T =0 − T Tˆt = 0 = tUB T − T LB LB

(22)

where T LB = 290K ,T UB = 340K

The model equations are solved using gPROMS custom modeling system (ProcessBuilder ver. 1.0.0) with nonlinear sequential quadratic programming (NLPSQP) optimization solver (optimization tolerance = 0.001). The spatial domain is discretized using centered finite difference method (CFDM) of second order with 10 points for the optimizations.

4. OPTIMIZATION RESULTS As shown in Table 3 indicating the basic constraints for the current optimization work, the optimization decision variables are the packing bed length, the contact time, the final Step ( 04) Step (10) Step( 06) Step( 09) = PEQ = PEQ equalized pressures ( PEQ , PEQ ) at pressure equalization steps, the 1 1 2 2

valve coefficients ( CVstep ( K ) ), and the variables ( φ ( z )t =0 ) at the beginning of the cycle for the CSS determination. The final equalized pressures and valve coefficients are decided to meet the mole balances of each interacting bed and all the optimal values of equalized pressures and valve coefficients are shown in Supporting Information, Tables S4 – S7. The following objective function (Obj) to be minimized and the basic constraints of Table 3 are adopted for all the present optimization works unless addressed. 12


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Obj = 105 ε

(23)

It would be good that the effects of the variables on the purity and recovery of methane are kept in mind for the analysis of the optimization results as shown in Figure 2. That is, the methane purity is proportional to the packing bed length, the bed inside diameter, the contact time, and the adsorption pressure, while it is inversely proportional to the regeneration purge pressure, the temperature, adsorption step time, and feed flow rate.5 The methane recovery shows the opposite trends to the methane purity. 5 4.1. Optimization Model for 10Nm3/h Scale Pilot Plants. Basic design conditions of 10Nm3/h scale (feed flow) VPSA process are efficiently attained through the optimizations. To gain the high purity methane product, the three cases (Cases 1a – 1c) are optimized, and the constraints for the optimization are same as the basic constraints as shown in Table 3 except for the following constraints for the Cases 1a – 1c. Purity CH 4 ≥ 0.97 , Recovey CH 4 ≥ 0.90 for Case 1a

(24.1)-(24.2)

PurityCH 4 ≥ 0.975 , RecoveyCH 4 ≥ 0.80 for Case 1b

(25.1)-(25.2)

PurityCH 4 ≥ 0.976 , Recovey CH 4 ≥ 0.75 , and

Lbed = 1.12m for Case 1c

(26.1)-(26.3)

The optimization results of the Cases 1a – 1c are shown in Table 4. As can be seen in Table 4, the highest methane purity (Case 1c) is obtained when the methane recovery is the lowest and the contact time is the longest. The optimal design condition were validated by PSA vendor 13



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company, GSA Co., Ltd in Korea, and the design condition of Case 1c was chosen as the specification of the VPSA pilot plant constructed at the CBM test site to get the highest purity of methane from CBM gas. In order to prove the current simulation model can be directly applied to scale up designs, simulation and optimization results are compared with the real pilot plant operation data. Table 5 lists the operating conditions for three operating scenarios of the selected pilot plant (Case 1c), and Table 6 describes the results of simulation and optimization agree well with the actual pilot plant operation data. In Table 6, there are very slight differences between the simulations and the optimizations because the simulation results are the values at almost CSS, i.e., CSScheck ≤ 10-3, and the optimization results are the values at perfect CSS, that is, CSScheck values in the optimizations are smaller than those in the simulations. Thus, the scale up design can be achieved by the current cyclic simulation and optimization approach which can efficiently get the perfect cyclic steady state. By adopting the Case 1a as a base case, the following three types of optimizations are investigated: 1) The optimizations are done at three different temperature conditions, that is, 293.15K for Case 1a(=Case 2a), 303.15K for Case 2b, and 313.15K for Case 2c. The optimization results are shown in Table 7. As to the increases of temperature, the methane purity is reduced and the methane recovery increases, while satisfying the performance specifications of methane purity (≥ 97%) and recovery (≥ 90%). The packing bed length increases to meet the methane purity specification by counteracting the effect of the decrease of the bed inside diameter and the contact time according to the increase of the temperature.

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2) The optimizations are carried out at three different adsorption step time, that is, 120 s for Case 3a, 140s for Case 3b(=Case 1a), and 160s for Case 3c. Here the second operating Step( 02) Step ( 05) step time ( t AD ) of the adsorption time must be same as the repose time ( t RE ), Step (11) Step ( 08) vacuum purge time ( t PG ), and the first pressurization time ( t PR ) for the continuous

purified methane production as described in Table 1. Thus, the following constraints must be satisfied. Step ( 02 ) Step ( 05 ) Step ( 08 ) Step (11) t AD = t RE = t PG = t PR

(27)

Step( 02) Here, it must be noted that the term, t AD , is a part of the total adsorption time Steps( 01− 03) ) as shown in Table 1. That is to say, ( t AD

Steps( 01− 03) Step( 01) Step( 02) Step( 03) t AD = t AD + t AD + t AD

(28)

As the results of the optimizations, Table 8 explains the shorter adsorption time results in the higher methane purity, the lower methane recovery, the longer contact time, a little longer bed length, and the slight bigger bed diameter. 3) The optimizations are conducted by adding more variables such as the adsorption step time, adsorption pressure, bed length, and purge pressure to the optimization decision variable set. Thus, the constraints, i.e., eqs (29) – (32), are applied to Cases 4a – 4c. Steps( 01− 03) 100s ≤ t AD ≤ 180s

(29)

Steps( 01− 03) 9 ×105 Pa ≤ PAD ≤ 13 ×105 Pa

(30) 15



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1.0m ≤ Lbed ≤ 1.5m ,

(31)

0.75s ≤ tcont ≤ 3.5s

(32)

Then the following constraints are also employed for Cases 4a – 4c. PurityCH 4 ≥ 0.975 , RecoveyCH 4 ≥ 0.9 for Case 4a

(33.1)-(33.2)

PurityCH 4 ≥ 0.9755 , RecoveyCH 4 ≥ 0.88 for Case 4b

(34.1)-(34.2)

Purity CH 4 ≥ 0.976 , Recovey CH 4 ≥ 0.85 , Step ( 08 ) 0.05 × 105 Pa ≤ PPG ≤ 0.1 × 105 Pa for Case 4c

(35.1)-(35.3)

To get the higher methane purity, longer contact time is required as described in Table 9. However, to minimize the sacrifice of the methane recovery level when the methane purity increases in VPSA processes, the adsorption step time increases, and adsorption pressure and bed length decrease. It can also be observed that the much longer contact time of Case 4c – i.e., almost twice of that of Case 4a – mitigates the effect of a little lower adsorption pressure on the methane purity in Case 4c than that in Case 4a. Compared to the results of Case 1c in Table 4, the methane recovery of Case 4c is much higher than that of Case 1c, and the methane purity of Case 4c is similar to that of Case 1c due to 1) the increase of the adsorption step time, 2) the decrease of the adsorption pressure and packing bed length, 3) bigger inside diameter, and 4) lower vacuum purge pressure as shown in Table 9.

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In addition, three more optimizations with different objective functions (equations (3638)) are performed using the constraints of Cases 4a – 4c.

Obj = 10 5 ε + Powersp,ave,Norm

for Cases 5a – 5c

Obj = 10 5 ε − PurityCH4 − RecoveryCH4 for Cases 6a – 6c

(36) (37)

Obj = 10 5 ε − PurityCH4 − RecoveryCH4 + Powersp,ave,Norm for Cases 7a – 7c

(38)

Vacuum purge pressure (PPG) constraint (equation 35.3) of Case 4c is also applied to Cases 5a – 7c. The normalized average specific power (Powersp,ave,Norm) is calculated by the following equations (39) – (43).

∫

t AD

0

Powercomp ,ave =

∫

t PG

0

PowerVP ,ave =

Powersp ,ave, =

  γ   PAD n& RT   feed z =0 γ − 1   P  atm    t AD

  

γ −1 γ

  − 1dt  

γ −1    γ     P γ n& RT z =0    feed z =0 γ − 1   P  − 1dt   PG      t PG

Powercomp,ave + PowerVP,ave n& prod

Powersp ,ave, Norm =

(39)

(40)

(41)

Powersp ,ave − Power LB

(42)

Power UB − Power LB

Power LB = 0; PowerUB = 105

(43)

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The results of the additional three types of optimizations (Cases 5a – 5c, Cases 6a – 6c, and Cases 7a – 7c) are shown in Tables S8 – S10 of Supporting Information. It is observed from the results (Tables S8 – S10) that 1) the enhancement of the methane recovery can be achieved by lowering the adsorption pressure and increasing the adsorption time, 2) the adsorption pressures (PAD) of Cases 5a – 5c and Cases 7a – 7c hit the lower bound to save the specific power consumption, and 3) the contact time (tcont) increases and vacuum purge pressure (PPG) is reduced to improve the methane purity.

4.2. Optimization Model for Scale up Design up to Large Commercial Size. To perform the scale up designs up to a large commercial scale VPSA plant, five optimization cases are achieved from 200Nm3/h through 5000Nm3/h scale in the feed flow aspect. To verify the current VPSA optimization quality based on dynamic simulation for the scale up design, this study takes following steps 1) The PSA vendor company, GSA Co., Ltd. in Korea, provides the packing bed length according to the five scales (200, 500, 1000, 2500, 5000Nm3/h in feed flow) of the 4-bed 12-operating step VPSA process to get the high methane purity (≥ 97%) and recovery (≥ 90%). 2) The optimizations are performed for each scale with the given bed lengths to find the proper design specification and to satisfy the desired performances, that is, the packing bed size, methane purity (≥ 97%), and methane recovery (≥ 90%). 3) The optimization results are compared with the vendor data.

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The vendor suggested packing bed lengths are 2.12m, 2.2m, 1.95m, 2.16m, 2.125m for 200Nm3/h, 500Nm3/h, 1000Nm3/h, 2500Nm3/h, and 5000Nm3/h respectively. The basic constraints in Table 3 except the contact time, bed diameter, bed length, and the temperature are adopted. In other words, the feed gas temperature, ambient temperature, and bed wall temperature are assumed to be 313.15K and the bed lengths are fixed as the vendor given values for each scale. The constraints of the bed inside diameter and the contact time are the followings:

0m ≤ Dinside ≤ 2.5m , 0.75s ≤ tcont ≤ 2.63s

(44.1)-(44.2)

As can be seen from Table 10, the scale up design optimizations are successfully carried out and validated very well. Because the relative mean errors are very small (less than 5%) in Table 10, the VPSA optimization model is excellently accurate and robust. The novel PSA optimization model,15 adopting the state of the art simulation model5 as equality constraints, makes it possible to perform the efficient scale up design instead of doing several simulations.

5. CONCLUSIONS This paper reports the scale up designs of VPSA processes are successfully performed by using the simulation and optimization technology. The total CPU times for the current optimizations with the objective function (Obj = 105ε) range from 200.773s through 813.374s which are very fast computations of the highly nonlinear rigorous dynamic VPSA model. 19



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However, it may take longer total CPU time (up to 3063.83s) to solve the optimization problems with the objective functions including methane purity and recovery, or normalized average specific power as shown in Supporting Information (Tables S8 – S10). The basic design conditions of 10Nm3/h scale (feed flow) VPSA process are found through the optimizations as shown in Table 4, and the conditions (Case 1c) that can produce the highest methane purity was selected for the design specifications of the pilot plant at the CBM test site. The effects of the temperature condition, adsorption time, and operating pressures are also observed as indicated in Table 7, Table 8 and Table 9. That is, methane recovery is proportional to the temperature and the adsorption step time, and is inversely proportional to the contact time, the bed length, bed inside diameter, and the adsorption pressure, while methane purity shows opposite trends to methane recovery. Though the bed length is generally inversely proportional to methane recovery as reported in the Ko’s paper (2016)5, the high temperature condition may allow a little bit longer bed length in obtaining the high methane recovery as depicted in Table 7. This means that the effect of temperature on the methane recovery can mitigate that of the bed length. Table 9 shows the longer adsorption step time, the lower adsorption pressure and shorter bed length are obtained as the optimal values in order to maintain the methane recovery level to be above the lower bounds as well as to satisfy the methane purity constraints. So the methane recovery of Case 4c (85%) can be much higher than that of Case 1c (78.56%). After the investigation of 10Nm3/h scale process optimizations, the optimizations for scale up design up to commercial large size VPSA process are performed. As described in Table 10, the proper bed sizes for each feed flow scale are found by the optimizations and validated very well. From the simulation study of the former paper (Ko, 2016)5 which was verified from the laboratory scale (~0.5Nm3/h) to pilot plant scale (100Nm3/h), this work extends to 20


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the optimization based scale up design study up to the large commercial scale (5000Nm3/h) VPSA plant. Thus it is proved that the current simulation model for the optimizations is successfully applied to 10,000 times scale up design. Though the trend of these optimization results may be slightly changed depending on starting points for optimizations due to the nonlinear optimization characteristics, the current optimization results can provide a good guide for the basic design of VPSA processes. As a conclusion, this work shows the strong possibility that the basic scale up design of PSA processes can be achieved not fully depending on the real plant experiences but relying on the current novel mathematical simulation and optimization technology.

Supporting Information Operating Step of Four-Bed VPSA, Adsorption Isotherms of methane, carbon dioxide, and nitrogen (Ko, 2016)5, Pressure Profiles of Pilot Plant Operation Data and Simulation Result (Operating Steps and Times are shown in Table S2), Comparison of Redlich-Kwong (RK) Equation of State with Ideal Gas Law in the Simulation5 (laboratory-scale & pilot-scale), Operating Step of Pilot Plant Operation, Comparison of Redlich-Kwong (RK) Equation of State with Ideal Gas Law in the Simulation (laboratory-scale & pilot-scale), Operating Step of Pilot Plant Operation, Comparison of nonadiabatic and adiabatic condition (laboratoryscale & pilot-scale), Optimal values of PEQ and CV in Cases 1a – 1c (different constraints of methane purity and recovery at 293.15K ( T feed = Tatm = Twall ), Optimal values of PEQ and CV in Cases 2a – 2c (different temperatures ( T feed = Tatm = Twall )), Optimal values of PEQ and CV in Cases 3a – 3c (different adsorption times at 293.15K ( T feed = Tatm = Twall )), Optimal values of PEQ and CV in Cases 4a – 4c at 293.15K ( T feed = Tatm = Twall ), Optimization Results of Cases

5a – 5c (tAD, PAD, and PPG are added to decision variable set at 293.15K ( T feed = Tatm = Twall ) and Obj = 10 5 ε + Powersp,ave,Norm ), Optimization Results of Cases 6a – 6c (tAD, PAD, and PPG 21



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are added to decision variable set at 293.15K ( T feed = Tatm = Twall ) and Obj = 10 5 ε − PurityCH4 − RecoveryCH4 ), Optimization Results of Cases 7a – 7c (tAD, PAD, and PPG are added to decision variable set at 293.15K ( T feed = Tatm = Twall ) and Obj = 10 5 ε − PurityCH4 − RecoveryCH4 + Powersp,ave, Norm )

This information is available free of charge via the internet at http://pubs.acs.org.

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AUTHOR INFORMATION Corresponding Author * Tel: +82 2 2154 6171. E-mail: [email protected] or [email protected]

Notes The author declares no competing financial interest.

ACKNOWLEDGEMENT This research has been co-funded by Korea Evaluation Institute of Industrial Technology (KEIT) under the Ministry of Trade, Industry & Energy (MOTIE) of Korea government and GS E&C. The author acknowledges KyungHyun Min, vice president of GSA Co., Ltd. in Korea, for the consultation and discussions.

NOMENCLATURE A

cross section area within bed [m2]

Avoid

void cross section area within bed [m2]

bi

Langmuir constant [1/bar] as a function of temperature

b0 ,i

Langmuir isotherm parameters [1/bar]

23



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C

total concentration [mole/m3]

Ci

concentration of component i [mole/m3]

C pg

heat capacity of gas [J/kg/K]

C ps

heat capacity of adsorbent [J/kg/K]

(K ) CVStep , ST

valve coefficient of step K at the step ST, here ST is step number [m3.5/kg0.5]

(K ) CVStep , EQ

valve coefficient of step K (pressure equalization) at the product end, here K is 04, 06, 09, or 10 [m3.5/kg0.5]

( 07 ) CVStep , BD

valve coefficient of step 07 (blowdown) at the feed end [m3.5/kg0.5]

08 CVStep , PG

valve coefficient of step 08 (purge) at the feed end [m3.5/kg0.5]

(11 − 12 ) CVSteps , PR

valve coefficient of steps 11-12 (pressurization) at the product end [m3.5/kg0.5]

Cw

heat capacity of bed wall [J/kg/K]

CSScheck

criterion variable for the CSS determination

CSS cycle

total number of cycles to reach CSS

Dax

axial dispersion coefficient [m2/s]

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D E ,i

isotherm parameters [K]

Dinside

bed inside diameter [m]

dp

adsorbent particle diameter [m]

∆H i

isosteric heat of adsorption [J/mole]

hInside

heat transfer coefficient of inside the bed [J/m2/s/K]

hOutside

heat transfer coefficient of outside the bed [J/m2/s/K]

i

gas component number denoting CH4, CO2, and N2

j

j = 1 when forward finite difference method (FFDM) or backward finite

difference method (BFDM) is used for axial discretization; j = 2 when centered finite difference method (CFDM) is used for axial

discretization.

ki

mass transfer coefficient of linear driving force (LDF) model of component i [1/s]

KL

effective axial thermal conductivity [J/m/s/K]

L

normalized bed length

Lbed

packing bed length [m]

25



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n

total mole [mole]

n&

molar flow rate [mole/s]

n& prod

product molar flow rate [mole/s]

nc

number of component

ND

number of discretization in finite difference method as to the bed axial domain

Obj

objective function which is minimized during the optimization

P

total pressure [Pa]

Pi

partial pressure [Pa]

Patm

atmospheric pressure [Pa]

Step (K ) PAD

adsorption pressure during the step K [Pa]

Step (K ) PEQ

equalized pressure of step K [Pa]

Step (K ) PBD

blowdown pressure of step K [Pa]

Step ( K ) PPG

vacuum purge pressure of step K [Pa]

PRegen

regeneration pressure, for example, PBD or PPG [Pa]

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P Step (K )

pressure of step K [Pa]

P Step ( K ), LB

lower bound of pressure at step K [Pa]

P Step ( K ),UB

upper bound of pressure at step K [Pa]

Pˆ

normalized pressure

Powercomp , ave

theoretical compressor power consumption [J/s]

PowerVP ,ave

theoretical vacuum pump power consumption [J/s]

Powersp ,ave,

theoretical total specific power consumption [J/s]

Powersp ,ave , Norm

normalized theoretical total specific power consumption [J/s]

PurityCH 4

average methane purity

Recovey CH 4

average methane recovery

R

universal gas constant [J/mole/K]

qi

adsorbed amount of component i [mole/g]

qi*

equilibrium amount adsorbed of component i [mole/g]

qs

equilibrium parameter for extended Langmuir isotherm [mole/g]

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q sa ,i

Langmuir isotherm parameters [mole/g]

q sb ,i

Langmuir isotherm parameters [mole K/g]

Rbed , Inside

inside radius of the bed [m]

Rbed ,Outside

outside radius of the bed [m]

RME

relative mean error [%]

Step (K ) t AD

operating time of step K (adsorption) [s]

Step ( K ) t EQ

operating time of step K (pressure equalization) [s]

Step ( K ) tBD

operating time of step K (blowdown) [s]

Step ( K ) tPR

operating time of step K (pressurization) [s]

t

time [s]

tcont

contact time [s]

t cycle

cycle time [s]

tsim

simulation time [s]

T

gas temperature [K]

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Tamb

ambient temperature [K]

Tfeed

feed gas temperature [K]

Twall

bed wall temperature [K]

T LB

lower bound of gas temperature [K]

TUB

upper bound of gas temperature [K]

Tˆ

normalized gas temperature

uI

interstitial gas velocity [m/s]

u Normal , feed

feed velocity at Normal condition [m/s]

V&Normal , feed

feed volume flow rate at normal condition [m3/s]

yi

mole fraction of component i

yiStep (K )

mole fraction of component i at step K

(K ) yiStep , ave

average mole fraction of component i at step K

z

normalized axial distance in bed from the feed inlet

Z

compressibility factor

29



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Greek Letters

µ

gas viscosity [kg/m/s]

ε bed

bed void

εt

total bed void fraction

ρbed

bed density [kg/m3]

ρg

gas density [kg/m3]

ρs

solid density [kg/m3]

ρw

wall density [kg/m3]

φ (z )t =0

variable standing for Tˆ , q, or y at the beginning of the cycle and the each discretization point of the spatial domain, z

φ (z )t =t

cycle

variable standing for Tˆ , q, or y at the end of the cycle and the each discretization point of the spatial domain, z

ε

nonnegative variable to be minimized during the optimization

REFERENCES (1) Ruthven, D. M.; Farooq, S.; Knaebel, K.S. Pressure Swing Adsorption; VCH publishers: New York, 1994. 30


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(2) Lea, J. F.; Nickens, H. V.; Wells, M. R. Gas Well Deliquification, 2nd ed.; Gulf Publishing: Burlington, 2008. (3) Sircar, S. Basic Research needs for the Design of Adsorptive Gas Separation Processes. Ind. Eng. Chem. Res. 2006, 45 (16), 5435. (4) Vetukuri, S. R. R.; Biegler, L. T.; Walther, A. An Inexact Trust-Region Algorithm for the Optimization of Periodic Adsorption Processes. Ind. Eng. Chem. Res. 2010, 49 (23), 12004. (5) Ko, D. Development of a Simulation Model for the Vacuum Pressure Swing Adsorption Process To Sequester Carbon Dioxide from Coalbed Methane. Ind. Eng. Chem. Res. 2016, 55 (4), 1013. (6) Smith, O. J.; Westerberg, A. W. Mixed-Integer Programming for Pressure Swing Adsorption Cycle Scheduling. Chem. Eng. Sci. 1990, 45, 2833. (7) Smith, O. J.; Westerberg, A. W. The Optimal Design of Pressure Swing Adsorption Systems. Chem. Eng. Sci. 1991, 46, 2967. (8) Nilchan, S.; Pantelides, C. C. On the Optimisation of Periodic Adsorption Processes. Adsorption 1998, 4, 113. (9) Ding, Y.; T Croft, D.; LeVan, M. Periodic States of Adsorption Cycles IV. Direct Optimization. Chem. Eng. Sci. 2002, 57, 4521. (10) Cruz, P.; Santos, J.; Magalhaes, F.; Mendes, A. Cyclic Adsorption Separation Processes: Analysis Strategy and Optimization Procedure. Chem. Eng. Sci. 2003, 58, 3143. (11) Cruz, P.; Magalhaes, F.; Mendes, A. On the Optimization of Cyclic Adsorption

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Separation Processes. AIChE J. 2005, 51, 1377. (12) Jiang, L.; Biegler, L. T.; Fox, V. G. Simulation and Optimization of Pressure-Swing Adsorption Systems for Air Separation. AIChE J. 2003, 49, 1140. (13) Jiang, L.; Biegler, L. T.; Fox, V. G. Simulation and Optimal Design of Multiple-Bed Pressure Swing Adsorption Systems. AIChE J. 2004, 50, 2904. (14) Ko, D.; Siriwardane, R.; Biegler, L. T. Optimization of Pressure Swing Adsorption Process using Zeolite 13X for CO 2 Sequestration. Ind. Eng. Chem. Res. 2003, 42, 339. (15) Ko, D.; Siriwardane, R.; Biegler, L. T. Optimization of Pressure Swing Adsorption and Fractionated Vacuum Pressure Swing Adsorption Processes for CO2 Capture. Ind. Eng. Chem. Res. 2005, 44, 8084. (16) Agarwal, A.; Biegler, L. T.; Zitney, S. E. Simulation and Optimization of Pressure Swing Adsorption Systems Using Reduced Order Modeling. Ind. Eng. Chem. Res. 2009, 48, 2327. (17) Agarwal, A.; Biegler, L. T.; Zitney, S. E. A Superstructure-Based Optimal Synthesis of PSA Cycles for Post-Combustion CO2 Capture. AIChE J. 2010, 56, 1813. (18) Khajuria, H.; Pistikopoulos, E. N. Optimization and Control of Pressure Swing Adsorption Processes under Uncertainty. AIChE J. 2013, 59 (1), 120. (19) Nikolic´, D.; Kikkinides, E. S.; Georgiadis, M. C. Optimization of Multibed Pressure Swing Adsorption Processes. Ind. Eng. Chem. Res. 2009, 48, 5388. (20) Ko D.; Moon I. Multiobjective Optimization of Cyclic Adsorption Processes. Ind. Eng. Chem. Res. 2002, 41, 93.

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(21) Sankararao B.; Gupta S. Multi-Objective Optimization of Pressure Swing Adsorbers for Air Separation. Ind. Eng. Chem. Res. 2007, 46 (11), 3751. (22) Fiandaca G.; Fraga E.; Brandani S. A Multi-Objective Genetic Algorithm for the Design of Pressure Swing Adsorption. Eng. Optimiz. 2009, 41 (9), 833. (23) Haghpanah R.; Majumder A.; Nilam R.; Rajendran A.; Farooq S.; Karimi I. A.; Amanullah M. Multi-Objective Optimization of a 4-Step Adsorption Process for Postcombustion CO2 Capture Using Finite Volume Technique. Ind. Eng. Chem. Res. 2013, 52 (11), 4249. (24) Haghpanah, R.; Nilam, R.; Rajendran, A.; Farooq S.; Karimi, I. A. Cycle Synthesis and Optimization of a VSA Process for Postcombustion CO2 Capture, AIChE J. 2013, 59 (12), 4735. (25) Olajossy, A.; Gawdzik, A.; Budner, Z.; Dula, J. Methane Separation from Coal Mine Methane Gas by Vacuum Pressure Swing Adsorption. Chem. Eng. Res. Des. 2003, 81 (4), 474. (26) Olajossy, A. Effective Recovery of CH4 from Coal Mine Methane Gas by Vacuum Pressure Swing Adsorption: A Pilot Scale Case Study. Chem. Eng. Sci. 2013, 1 (4), 46. (27) Gomes, V. G.; Hassan, M. M. Coalseam Methane Recovery by Vacuum Swing Adsorption. Sep. Purif. Technol. 2001, 24, 189. (28) Delgado, J. A.; Rodrigues, A. E. Analysis of the Boundary Conditions for the Simulation of the Pressure Equalization Step in PSA Cycles. Chem. Eng. Sci. 2008, 63, 4452. (29) Kumar R.; Fox V. G.; Hartzog D. G.; Larson R. E.; Chen Y. C.; Houghton P. A.; Naheiri 33



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T. A Versatile Process Simulator for Adsorptive Separations. Chem Eng Sci. 1994, 49 (18), 3115. (30) Patel, M. Optimising Adsorption Process Design and Operation, AIChE Webinar, October 29, 2014. (31) Edgar, T. F.; Himmelblau, D. M.; Lasdon, L. S. Optimization of Chemical Processes, 2nd ed.; McGraw-Hill: New York, 2001.

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Figure 1. Four-bed Twelve-Operating Step VPSA Process Figure 2. Effects of Design and Operation Variables on the Purity and Recovery of the Purified Product

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Figure 1. Four-bed Twelve-Operating Step VPSA Process

Figure 2. Effects of Design and Operation Variables on the Purity and Recovery of the Purified Product

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Table 1. Four bed twelve operating step sequence and times5 Table 2. Summary of PSA modeling approaches Table 3. Basic Optimization Constraints when feed flow rate is 10Nm3/h Table 4. Optimization results of Cases 1a – 1c (different constraints of methane purity and recovery at 293.15K ( T feed = Tatm = Twall )) Table 5. Conditions for three Operation Scenarios of the Selected Pilot Plant Table 6. Comparison of Pilot Plant Operation Data with Cyclic Simulations and Optimization for CSS determination Table 7. Optimization Results of Cases 2a – 2c (different temperatures ( T feed = Tatm = Twall ) when feed flow rate is 10Nm3/h) Table 8. Optimization Results of Cases 3a – 3c (different adsorption time at 293.15K ( T feed = Tatm = Twall )) Table 9. Optimization Results of Cases 4a – 4c at 293.15K ( T feed = Tatm = Twall ) Table 10. Optimization Results for VPSA scale up design at 313.15K ( T feed = Tatm = Twall )

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Table 1. Four bed twelve operating step sequence and times5 Step No.

1

2

3

4

5

6

7

8

9

10

11

12

Time (s)

10

120

10

10

120

10

10

120

10

10

120

10

Bed 1

AD

AD

AD

EQ1

RE

EQ2

BD

PG

EQ2

EQ1

PR

PR

Bed 2

BD

PG

EQ2

EQ1

PR

PR

AD

AD

AD

EQ1

RE

EQ2

Bed 3

EQ1

PR

PR

AD

AD

AD

EQ1

RE

EQ2

BD

PG

EQ2

Bed 4

EQ1

RE

EQ2

BD

PG

EQ2

EQ1

PR

PR

AD

AD

AD

AD: adsorption, EQ: pressure equalization, RE: repose, BD: blow down, PG: vacuum purge, PR: pressurization

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Table 2. Summary of PSA modeling approaches Variable

Equation for calculation of variables Extended Langmuir isotherm: qi* =

* i

q

qs ,i bi Pi (1)

nc

1 + ∑ bi Pi i =1

here, qs , i = qsa , i +

qsb , i D  , bi = b0, i exp E , i  T  T 

(2.1)-(2.2)

∂qi = ki qi* − qi ∂t

(

)

qi

Linear driving force (LDF) model equation:

C

Equation of state (EOS): P = ZCRT Component concentration: Ci = yiC Mole fraction balance5:  ∂2 y ∂(1/ T ) ∂yi ∂y ∂P ∂(1/ Z ) ∂yi   − Dax  2i + 2T + 2(1/ P ) i + 2Z ∂z ∂z ∂z ∂z ∂z ∂z   ∂z

Ci

yi

+ uI

(3) (4) (5)

(6)

∂yi ∂yi ρ s RTZ (1 − ε bed )  ∂qi ∂q  + + − yi ∑ i  = 0  ∂z ∂t P ε bed  ∂t i =1 ∂t  nc

Ergun equation when a interstitial gas velocity ( uI ) is used:

P

uI

T

2 ( ∂P 1 − ε bed ) µ 1 − ε bed ρ g − = 150 u + 1.75 u u 2 2 I ε bed d p I I ∂z dp ε bed

Interstitial gas velocity equation [24] (Ko, 2016): Pu I Avoid = Zn& RT Energy balance of the gas within adsorption bed: (ε t ρ g C pg + ρ bed C ps ) ∂∂Tt + ρ g C pg ε bed u I ∂∂Tz nc 2 hInside ∂ 2T ∂q − K L 2 − ρ bed ∑ ∆H i i + (T − Twall ) = 0 ∂z ∂t Rbed , Inside i =1 Energy balance of the adsorption bed wall: (ρ wCw )π Rbed ,Outside 2 − Rbed , Inside 2 ∂Twall ∂t = 2πRbed , Inside hInside (T − Twall ) − 2πRbed ,Outside hOutside (Twall − Tamb ) Mole balance5:

(

n&

)

 L ∂n + (1 − ε bed ) j bed ∂t  ND

nc  ∂q ∂n&  L  A ρ s ∑ i +   j bed ∂ t  ∂z  N D i =1 

  = 0 

39


(7)

(8)

(9)

(10)

(11)


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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 40 of 48

Table 3. Basic Optimization Constraints when feed flow rate is 10Nm3/h Variable

LB

UB

PurityCH 4 (%)

97.00

100.00

Recovey CH 4 (%)

90.00

100.00

Lbed (m)

1.0

1.3

tcont (s)

0.75

2.7

Steps( 01− 03) PAD (Pa)

11×105

11×105

Step ( 04 ) PEQ (Pa) 1

3×105

11×105

Step ( 06 ) PEQ (Pa) 2

1.1×105

7×105

Step ( 07 ) PBD (Pa)

1×105

1×105

Step ( 08 ) PPG (Pa)

0.1×105

0.1×105

CVstep ( K ) (m3.5/kg0.5)

10-10

0.001

−ε

ε

Tˆt = 0 , qCO 2 , t = 0 , q N 2 , t = 0 , yi , t = 0

0

1

q CH 4 , t = 0

0

10

T feed = Tatm = Twall (K)

293.15

293.15

DInside (m)

0.0

2.0

φ (z )t =0 − φ (z )t =t

cycle

LB = lower bound, UB = upper bound CVstep ( K ) stands for

( 04) step( 06) step( 07) step( 08) step( 09) step(10) steps(11−12) CVstep , EQ1 , CV , EQ 2 , CV , BD , CV , PG , CV ,EQ 2 , CV , EQ1 , or CV , PR φ stands for Tˆ , q, or y at each discretization point ε is nonnegative variable to be minimized during the optimization i denotes CH4, CO2, or N2 Operating step times in Table 1 Feed pressure (Pfeed) is same as adsorption pressure (PAD) Model equations (1) – (11) in Table 2

40


Page 41 of 48

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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 4. Optimization results of Cases 1a – 1c (different constraints of methane purity and recovery at 293.15K ( T feed = Tatm = Twall )) Case 1a PurityCH 4 ≥ 97%

Case 1b Purity CH 4 ≥ 97.5%

Case 1c Purity CH 4 ≥ 97.6%

RecoveyCH 4 ≥ 90%

RecoveyCH 4 ≥ 80%

RecoveyCH 4 ≥ 75%

PurityCH4 (%)

97.4101

97.5453

97.6033

Recovery CH4 (%)

91.7711

85.8561

78.5593

PurityCO2 (%)

0.1128

0.01079

6.5319×10-4

Recovery CO2 (%)

1.1630

0.1039

5.7538×10-5

Lbed (m)

1.1194

1.0011

1.12

tcont (s)

1.1409

1.8146

2.6399

Dinside (m) Specific Powerave (J/mol) Obj

0.09938

0.1325

0.1511

9243.155

9921.895

10887.728

0

0

0

Variable

482.121 405.915 Total CPU time (s) Computer specifications for the optimizations: Process: Intel® Core i5-4310M CPU @ 2.7GHz, RAM: 4 GB, 64 Bit OS.

41


200.773


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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 48

Table 5. Conditions for three Operation Scenarios of the Selected Pilot Plant SC1a

SC1b

SC1c

9.017332

8.5094

9.9599

PAD (bar)

8.2882

7.2087

7.8714

PPG (bar)

0.18025

0.15857

0.18868

93% : 7%

82% : 18%

72% : 28%

Case Feed Flow (Nm3/h)

Feed gas (yCH4 :yCO2) Lbed(m)

1.12

Dinside(m)

0.151

Operating Step Times

See Table S2

42


Page 43 of 48

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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 6. Comparison of Pilot Plant Operation Data with Cyclic Simulations and Optimizations for CSS determination Case

Variable

Experiment

Simulation

99.76

99.9839

0.2242

99.9831

0.2234

RecoveryCH4 (%)

80.8794

81.1595

0.3457

81.1551

0.3403

Product Flow (Nm3h-1)

6.7990

6.8233

0.3568

6.8230

0.3524

CSSCycle

N.A.

11

N.A.

N.A.

N.A.

CSScheck

N.A.

6.690×10-4

N.A.

1.114×10-5

N.A.

Total CPU Time (s)

N.A.

226.545

N.A

303.656

N.A.

PurityCH4 (%)

99.04

99.9800

0.9446

99.9790

0.9436

RecoveryCH4 (%)

81.06

82.1780

1.3698

82.1770

1.3686


5.7806

5.7615

0.3310

5.7615

0.3310

CSSCycle

N.A.

11

N.A.

N.A.

N.A.

CSScheck

N.A.

6.676×10-4

N.A.

2.926×10-5

N.A.

Total CPU Time (s)

N.A.

242.348

N.A

149.839

N.A.

PurityCH4 (%)

98.89

99.9118

1.0280

99.9119

1.0281

RecoveryCH4 (%)

86.9768

86.7932

0.2113

86.7943

0.2100


6.3072

6.2711

0.5740

6.2711

0.5740

CSSCycle

N.A.

12

N.A.

N.A.

N.A.

CSScheck

N.A.

3.120×10-4

N.A.

1.495×10-4

N.A.

Total CPU Time (s)

N.A.

282.362

N.A

272.612

N.A.

PurityCH4 (%)

SC1a

SC1b

SC1c

RME(%) Optimization RME(%)

[Note] CSSCycle = total number of cycles to reach the cyclic steady state (CSS). nc N D +1

nc −1 N D

ND

i =1 k =1

i =1 k = 2

k =2

CSS check = ∑ ∑ qi ,t =0,k − qi ,t =tcycle ,k + ∑∑ yi ,t =0,k − yi ,t =tcycle ,k + ∑ Tt =0,k − Tt =tcycle ,k

CSS is assumed to be satisfied when CSScheck

Optimization of Vacuum Pressure Swing Adsorption Processes To

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