A Systematic Simulation and Proposed Optimization of the Pressure

Article pubs.acs.org/IECR

A Systematic Simulation and Proposed Optimization of the Pressure Swing Adsorption Process for N2/CH4 Separation under External Disturbances Weina Sun,† Yuanhui Shen,† Donghui Zhang,*,† Huawei Yang,† and Hui Ma‡ †

Collaborative Innovation Center of Chemical Science and Engineering, School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China ‡ School of Architecture, Tianjin University, Tianjin 300072, China ABSTRACT: This work presents a detailed study of the systematic simulation, optimization, and control of a pressure swing adsorption (PSA) process that used activated carbon as adsorbent to recover CH4 from a gas mixture (70% N2/30% CH4). The state-of-the-art reduced space successive quadratic-programming (r-SQP) optimization algorithm is employed to find the optimal values of the decision variables with additional constraints. The best closed loop recovery obtained for the PSA system under consideration is around 98% with purity of 80%. The control strategy is based on the regulatory proportional−integral− derivative (PID) controller because of its practicability and stability. Additional constraints that guaranteed the flexibility and adaptability of the PID controller were imposed on the optimization-based PSA system. The ability of the control system to reject various disturbances was evaluated and compared with the open loop conditions. Results demonstrated that the welldesigned control system showed a wonderful performance in the presence of multiple disturbances.

1. INTRODUCTION In coal bed methane (CBM), N2 and CH4 are the main components and the concentration of methane usually lies between 20% and 35%, which means the CBM is unable to be utilized directly. In China, most CBM is discharged into the atmosphere, which is a waste of a resource and increases the greenhouse effect.1 The greenhouse warming potential of CH4 is 25 times higher than that of the well-known greenhouse gas carbon dioxide.2 From the standpoint of environmental protection, upgrading the low-concentration coal bed methane is extremely advantageous with the more stringent restriction on the emission amount of greenhouse gases such as carbon dioxide and methane. Meanwhile, methane has a lower cost than traditional fossil fuels such as gasoline or diesel, which can generate significant economic benefit3,4 in the face of the severe global energy situation. To remove the existing nitrogen, many unit operations of chemical engineering such as cryogenic distillation,5 membrane-based separation,6 and pressure swing adsorption (PSA)1,7−9 have been proven to be feasible. Among all the feasible unit operations, PSA proved itself to be the most desirable because of its low operating cost compared with cryogenic distillation and efficient automatic operation in contrast with membrane-based separation. The inventions of PSA occurred before the underlying theories behind it were fully understood. Since its practical application in the late 1950s, PSA technology has evidenced substantial growth in terms of scale, versatility, and complexity.10 The PSA process consists of multiple interactive beds filled with adsorbents and operates in a cyclic manner, which is called periodic operation. To simulate and optimize the PSA system, rigorous mathematical models consisting of coupled partial differential and algebraic equations (PDAEs) distributed over time and space that describe material, energy, and the momentum balances together with transport phenomena and © XXXX American Chemical Society

equilibrium equations have to be formulated to describe the highly nonlinear nature and dynamic behavior of the real PSA plants.11,12 In the last three decades, many mathematical models of adsorption beds with different complexities have been established and simulated. Owing to the tedious and timeconsuming solution procedure of the thousands of PDAEs of the whole PSA system, those not only accurate but also simplified models are desirable to decrease the essential computational time and accelerate optimization studies.13 In addition to its highly nonlinear and dynamic nature, the PSA system poses extra challenges because of its highly interactive characteristics among process variables, especially design parameters such as step times, pressure, temperature, gas velocity, and bed dimensions. Consequently, it is difficult to find the optimum conditions through experiments of trial and error, which is a waste of time and an irrational utilization of resources. A great deal of research concerning simulation and optimization of simplified model-based PSA processes has been reported in the literature. Nilchan proposed a complete discretization approach.14 In this approach, the space and the temporal domain are discretized simultaneously, which reduces the PDAEs-based model equations to a large set of nonlinear algebraic equations. The resulting optimization formulation is a rather small nonlinear programming (NLP) problem which is easy to solve using a standard NLP solver. Jiang et al. apply the direct determination approach proposed by Smith and Westerberg16 and Croft and LeVan17 and simultaneous tailored approach, a reduced space successive quadratic-programming Received: March 24, 2015 Revised: July 6, 2015 Accepted: July 9, 2015

A

DOI: 10.1021/acs.iecr.5b01862 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 1. Graphical overview of the three-bed PSA system.

product needs to be controlled through incorporating proper controllers. Simulation and control of the model-based chemical process has significant meaning for the practical operation in industry. During the last few decades, a number of studies have been made on the control of individual unit operations in the chemical industry, such as distillation and crystallization. For example, Sen et al. designed an efficient control system for the continuous production of active pharmaceutical ingredients which incorporated the control scheme of unit operations crystallization, filtration, drying and so on.28 Batch reactors control techniques had also been designed and assessed thoroughly to improve the performance.29 Singh et al. had designed a system-side proportional− integral−derivative (PID) control strategy and demonstrated the improved performance of manufacturing process with the controllers compared to the open loop simulations. They proposed another hybrid model predictive control (MPC)-PID control scheme for the continuous pharmaceutical tablet manufacturing process after about one year.30−32 Because of the dynamic and highly nonlinear nature accompanying the cyclic operation of the PSA process, the design of an effective controller for the PSA system is a challenging yet meaningful task. There are very few studies with respect to the design of the control system for industrial PSA operations.33−39 Blitzer et al. presented a process control scheme consisting of a nonlinear feedforward and a linear feedback control and verified the scheme using a rigorous simulation model of the PSA plant.33−35 Torre et al. also designed a model predictive controller for periodic adsorption process.37 Khajuria et al. proposed a model predictive control (MPC) for a PSA system which involved four beds separating 70% H2, 30% CH4 mixture into high purity hydrogen.38 To derive the proposed MPC controller, a rigorous and systematic framework is employed, and closed loop simulations were performed to test the performance of the corresponding MPC controller and to compare with conventional PID controller. Afterward, they extended the MPC controller to a two-bed, six-step PSA system with the same composition of feed gas to control product purity to the desired point, 99.99%. Bitzer set up a control scheme consisting of a nonlinear feedforward and a linear feedback control.39 Nevertheless, the feedforward control derived from a

(r-SQP) optimization strategy to three different nonisothermal single-bed industrial O2 vacuum swing adsorption (VSA) processes to compare the cyclic steady state (CSS) convergence speed with the time-consuming successive substitution method.15−17 Later, they extend their methodology to multiple bed systems. It approximated the computation of the enormous sensitivity matrix of the CSS equations by low-order parametrization, which may introduce inaccuracy into design and optimization results.18 Agarwal et al. presented a generic PSA superstructure and adopted the complete discretization methodology, which is capable of determining the optimal PSA cycle configurations and optimal sequence steps of the whole PSA process, and demonstrated its potential by applying to the not only precombustion but also postcombustion CO2 capture.19−21 To accelerate the CSS, they also adopted reduced order models (ROM), proper orthogonal decomposition (POD), and incorporated it into the trust-region-based optimization framework.22 Smith and Westerberg performed the optimization applying the r-SQP algorithm to find the optimal equalization stage configurations and the optimal operating conditions for different adsorbent types.23 Several cases of oxygen production from air by PSA and VSA processes adopting this proposed approach have been illustrated. Ko et al. used the r-SQP method to find the optimal values of the decision variables, such as flow rates, bed pressure, step times, and bed dimension.24 The PSA process has been studied frequently in recent years to capture precombustion and postcombustion CO2 to reduce the emission of the greenhouse gas. Dowling et al. proposed a systematic optimization-based formulation for the synthesis and design of novel PSA cycles for CO2 capture in integrated gasification combined cycle power plants applying a superstructure-based approach.25 Huang et al. studied the optimization of one major application of PSA process (hydrogen purification from the plasma reactor gas) by developing a rigorous PSA model and solving it using gPROMS (Process System Enterprise) software.26,27 Purity, recovery, and the overall productivity of the product are of utmost importance. Nevertheless, there are inevitably unknown disturbances and parameter uncertainties that may induce constraint violations and performance degradation in the actual industries. This means that the quality of the desired B


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Industrial & Engineering Chemistry Research Table 1. Schedule of the VPSA Processa time (s)

50

200

50

50

200

50

50

200

50

bed1 bed2 bed3

PR↑ VU1★↓ RP1★↑

AD↑ VU2★↓ RP2★↑

ED↑ ER↓ BD↓

RP1★↑ PR↑ VU1★↓

RP2★↑ AD↑ VU2★↓

BD↓ ED↑ ER↓

VU1★↓ RP1★↑ PR↑

VU2★↓ RP2★↑ AD↑

ER↓ BD↓ ED↑

a

PR, pressurization step; AD, adsorption step; ED, depressurization equalization step; RP1/RP2, replacement (heavy component purge) step; BD, blowdown step; VU1/VU2, evacuation step; ER, repressurization equalization step. ★: For the convenience of explanation in the following process control part, the RP and VU step is divided into RP1 and RP2, VU1 and VU2, respectively, which are the same step essentially.

Table 2. Model Equations of VPSA Process parameter component mass balance

expression 42−45

2

− εbDax overall mass balance

∂(vgc)

∂ yi ∂z 2

+

∂(vgci)

+ (εb + (1 − εb)εp)

∂z

+ (εb + (1 − εb)εp)

∂z

∂q ∂ci + ρp (1 − εb) i = 0 ∂t ∂t

∂q ∂c + ρp (1 − εb) =0 ∂t ∂t

(1)

(2)

N

energy balance

[(εb + (1 − εb)εp) ∑ ci(c pg, i − R ) + (1 − εb)ρp c ps + (1 − εb)ρp i=1 N

∑ qi(cpg,i − R)] ∂T i=1 N

∑ i=1

gas-phase momentum balance

41

− Langmuir isotherm

linear driving force model

∂qi ∂t

ΔHi + 2h

+ (1 − εb)ρp

T − Tw ∂P ∂ 2T − (εb + (1 − εb)εp) − kg 2 = 0 ∂t Rb ∂z

qm, i bi Pi

qi* = ∂qi

n

1 + ∑i = 1 biPi

=

15Dc, i (R p)2

,

⎛ −ΔHi ⎞ ⎟ bi = b0 exp⎜ ⎝ RT ⎠

(qi* − qi)

Dax = 0.73Dm +

z = 0:

⎛ ε D ⎞ εb⎜1 + 9.49 2vb Rm ⎟ ⎝ g p⎠

else,

,

∂yi , z ∂Tz = 0, =0 ∂z ∂z

εp

P(Dv ,A

Dk , iDm

τ Dk , i + Dm

(7a, 7b)

0.1013T1.75 Dm =

1/3

( +

1 MA

+

1 MB

)

Dv ,B1/3)2

(7c, 7d)

(8a)

(8b)

vg, z < 0, Tz = Tout , yi , z = yout, i

unidirection:

(4)

(5a, 5b)

Dc, i =

vg, z ≥ 0, Tz = Tin , yi , z = yin, i

z = Hb:

(3)

(6) vgR p

∂yi , z ∂Tz else, = 0, =0 ∂z ∂z

valve equation

∂z

i=1

⎛T ⎞ Dk , i = 48.5D p ⎜ ⎟, ⎝ Mi ⎠ boundary conditions

∑ cpg,i ∂T

(1 − εb)ρg 150μ(1 − εb)2 ∂P = vg + 1.75 |vg|vg 3 2 ∂z εb (2R p) 2R pεb3

∂t diffusion coefficients

∂t

N

+ vgρg

(8c)

(8d)

if Pin > Pout , then F = Cv(Pin − Pout)/106 else, then F = 0

bidirection:

F = Cv(Pin − Pout)/106

(9)

handling large-scale and complex PSA systems but also is more

reduction model which is the approximation of the original rigorous distributed parameter model, and the practicability of the control solution had not yet been fully demonstrated. In this paper, a systematic model-based simulation, optimization, and control of a PSA process for methane production is presented. A PDAEs-based model of the vacuum pressure swing adsorption (VPSA) process for separation of CH4 from a N2-rich mixture is considered with the feed composition of 30% CH4 and 70% N2. The state-of-the-art rSQP optimization strategy, which not only is capable of

robust and efficient than other optimization methodologies, was implemented successfully in gPROMS to maximize the product recovery with the necessary constraints. The fine designed control system is applied to the optimization-based PSA system with various disturbances, and the closed-loop performance is compared with the open loop simulations. C


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Industrial & Engineering Chemistry Research Table 3. Process Performance Indicators performance indicator

expression used

power consumption

if Pout > Pin , then dW =

Wcycle =

∫0

tcycle

purity

purityCH =

∫0

dW

tcycle

4

∫0

recoveryCH = 4

(11)

Foutyout,CH dt

tcycle

tcycle

∫0

productivity

Fout dt

(12)

Foutyout,CH dt 4

tcycle

Finyin,CH dt 4

3600(∫

0

productivityCH =

tBD

(13)

Foutyout,CH dt + ∫

0

4

tVU

Foutyout,CH dt ) 4

tcyclewads

4

wads = ρp (1 − εb)πHbR b

2. PSA MODEL AND SIMULATION 2.1. Model Assumptions. The PSA system usually comprises two or more adsorbent-filled beds interconnected to each other via a network of switch valves. During the adsorption stage, the feed gas is introduced to the bed and adsorbed substantially at high pressure. The beds are later regenerated by desorbing the remaining adsorbed gases at low pressure. In our study, we consider a three-bed and seven-step VPSA process. Figure 1 shows the flow diagram of the process, and the schedule is defined in Table 1.The feed gas passed through the adsorption bed after compression. A stream of N2 was withdrawn from the bed during the adsorption step (AD). Following the depressurization equalization step (ED) was the replacement step (RP1 and RP2), which used part of the product CH4 stored temporarily in gas_buffer2 and aimed to increase the concentration of CH4 and exclude N2 from the adsorption bed. The blowdown (BD) and evacuation (VU1 and VU1) steps fully regenerated the adsorbent and extracted the high-purity CH4 product. Finally, the repressurization equalization step (ER) and pressurization step (PR) were performed, which increased the pressure of the bed to make it ready for a new cycle. From the perspective of modeling, a VPSA model should be able to capture the dynamics of the system during the whole cyclic stages. A detailed, first-principles-based mathematical model of the PSA system is developed for this study in gPROMS. As the core part of the entire VPSA model, a mathematical model of the adsorption bed is described in Table 2, and listed below40−45 are some assumptions made for the model: (1) The gas phase behaves as an ideal gas. (2) There is no radial variation in gas concentration, temperature, and pressure. (3) Pressure drop along the bed is calculated by the Ergun equation.42 (4) There is thermal equilibrium between the gas and solid phases. (5) The porosity of the bed and adsorbent particle is uniform along the bed. (6) LDF approximation model with a single lumped mass transfer coefficient is used.

(10)

4

∫0

recovery

γ − 1/ γ ⎧ ⎫ ⎪ PoutVin γ ⎪⎛ Pout ⎞ ⎨⎜ − 1⎬ ⎟ ⎪ ηp γ − 1 ⎪ P ⎝ ⎠ ⎩ in ⎭

2

(14)

(15)

(7) Extended Langmuir nonisothermal models are used to describe the adsorption behaviors. Valve equation (eq 9) shows the calculations for mole flux variables which are used to predict the performance. Based on the mole flux variables above, purity and recovery of product methane and the other two commonly used performance variables are given in Table 3. Table 4 lists the Langmuir nonisothermal model parameters. In Tables 5 and 6, parameters for the adsorption bed model and the gas−solid system are defined.43 Table 4. Langmuir Nonisothermal Model Parameters (313.15 K) parameter46

N2

CH4

b0 qm ΔH

1.73 × 10−6 2.97 −17.082

1.17 × 10−6 3.47 −21.267

Table 5. Parameters for Adsorption Bed Model design parameter

value

Pfeed Tfeed Qfeed Hb Rb τ εb h

3 × 105 303.15 18 1.0 0.15 4.5 0.35 55.0

2.2. Cyclic Steady State Definition. CSS, which is the desired “steady” state in the whole PSA system, is an operation condition where all differential and algebraic variables have the same values at the beginning and the end of one cycle. It is the result of process variable accumulation and improvement by means of receiving final state variables from the previous cycle and providing those updated variables to the next cycle. Successive substitution and direct determination are two commonly used methods to define the CSS. Although the former may take more cycles than the direct determination approach to reach the CSS, successive substitution can provide D


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the two contradictory indicators of real-time PSA performance simultaneously in the optimization framework. In this study, an optimal PSA operational policy is desired which achieves the maximum value of recovery for the separation of 70% N2 and 30% CH4 feed mixture to methane product purity no less than 80%. The decision variables related to PSA process design, operation, and cycle scheduling should be selected carefully to obtain an optimal performance. The key process design variables considered here are valve CVs, while feed flow rate is included as an operational decision variable. It is generally known that dynamic optimization without any constraints can often lead to unrealistic physical behavior in which many of the process state variables violate their logical limits. To find the optimal operation conditions for the designed PSA system, adding the necessary operational constraints is as important as the selection of the objective and decision variables. The entire dynamic optimization problem formulated according to the above criterion is expressed as follows with the modeling parameters listed previously:

Table 6. Modeling Parameters for the Gas−Solid System parameter46

value

Cpg Cps dp Rp ρp DvCH4

0.735 0.887 1 × 10−9 6.0 × 10−4 525 24.5

DvN2

18.5

kg kLDF,CH4

0.242 1.282

kLDF,N2

3.020

μ εp

1.68 × 10−5 0.33

a global observation for the change of all process variables and has favorable stability and convergence. In this work, successive substitution is adopted to define the relaxed definition of CSS and is expressed as eCSS = |yt = Nt

cycle

− yt = (N + 1)t | ≤ εCSS cycle

Min − R s.t. F(y , y′ , q , t ) = 0

(16)

purityCH ≥ 80.00%

where y represents the significant variables (the separation performance indicators of the VPSA system, purity and recovery) used to define the CSS and εcss is a small positive value, which plays a critical role in the evaluation of the number of cycles required before reaching the CSS. In this work, the value of εcss is 10−5. 2.3. Discretization Method. In this study, the method of lines (MOL) and central finite difference method (CFDM) are employed to convert the PDAE system to a differential algebraic equation (DAE) system. The DAE system was dynamically simulated by intergrading over time in a DAE integrator, and the results of the CSS was transferred to the dynamic optimizer (gOPT) in gPROMS as initial values for all the variables. It is a well-known fact that most great changes in process variables usually occur in the initial few cycles. Therefore, a value of 40 cycles is selected as the time horizon for PSA dynamic optimization studies.

4

W ≤ 0.25 Pevacuation ≥ 10 kPa Pblowdown ≥ 101.3kPa Cv7 = Cv8 Cv7 = Cv9 q LB ≤ q ≤ q UB

(17)

Here, F is the DAE model from Table 2; y are the state variables; q are decision variables such as flow rates which are subject to the lower bounds (LB) and upper bounds (UB). Cv7 is the equalization valve constants of bed1, Cv8 the equalization valve constants of bed2, and Cv7 the equalization valve constants of bed3. For the PSA system shown in Figure 1, the resulting optimization configuration obtained by the employment of the approach mentioned above is shown in Table 7, whereas the corresponding computational statistics are outlined in Table 8. Note that for the three-bed PSA system under consideration, the total CPU time to solve the full optimization problem is around 64 h. Table 7 also depicts the optimal values of key degrees of freedom related to process design and operation of the PSA system. 3.2. Process Analysis. To further evaluate the design performance, dynamic simulations are performed with the optimal PSA system listed in Table 7. In the following discussion, this optimization configuration will be referred to as nominal PSA design. The spatial distribution of concentration of CH4 at the end of each processing step at the optimal operation configuration as well as initial condition are shown in Figures 3 and 4. Pressure evolution profiles obtained from the simulation results of optimal and initial conditions were also displayed to better understand the complexity of the PSA process and the intricate interaction of their operating parameters. As we can see from Figure 2, the optimum feed pressure of the whole process is lower than that at initial conditions. The gas-phase concentration of CH4 in the outlet side of the bed at the optimal condition experienced a

3. OPTIMIZATION 3.1. Formulation of the Optimization. The PSA operation is dynamic in nature, which means that it will never attain a true steady state, thus requiring computationally expensive simulations to be carried out for hundreds of cycles until cyclic steady state is achieved. In this work, the traditional dynamic optimization approach is considered to be more suitable for control study and is employed. The CSS is chosen as the initial values for the dynamic optimization, which is an excellent choice to speed up the convergence. Another advantage coming from the software is that it updates the standard solver of nonlinear optimization problems from SRQPD to NLPSQP. It is assumed that the structure of the PSA cycle and sequence of processing steps remains fixed for the optimization studies. The remainder of this paragraph is organized as follows: choice of the objective function, decision variables, and determination of the necessary constraints. Product purity is usually set by the customer requirements, while recovery is to be maximized at the specified purity levels. However, the two performance indicators generally vary toward the opposite direction with respect to significant decision variables. Therefore, in a PSA system, it is ideal to incorporate E


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slightly forward with the existence of the low-concentration region, which can be observed in Figure 3. This improvement was the result of the coordination of each processing step. Any mismatching or a little inappropriate change in the valve constant of a certain valve could lead to the degradation of the process performance. For example, a larger valve constant of the equalization valve can lead to the penetration of the adsorption front and product leaving from the outlet of the bed, resulting in the pollution of the other bed undergoing repressurization equalization step and the decrease of product recovery. One major purpose of the introduction of the equalization step is to utilize the energy from the high-pressure bed. A smaller valve constant of the same equalization valve will not efficiently utilize the energy of the impurity available at high pressure and degrade the product obtained at the regeneration stage. As a VPSA process for heavy component enrichment, the replacement step played a very important role in improving product purity. The flow rate of replacement gas under the optimal condition is 6.108018. It is very crucial to control the extent of the replacement to increase the concentration of CH4 along the bed and exclude N2 from the bed. Excessive replacement can lead to the loss of product and waste of compression energy. Nevertheless, deficient replacement is not enough to maintain the heavy component at the desired purity level. The blowdown step marked the beginning of the adsorbent regeneration stage. Blowdown and evacuation steps played the same role, with the only difference that blowdown does not consume any energy. The optimal pressure of blowdown and evacuation were almost at their lower bounds of the optimization constraints. This meant that the production of CH4 could be improved by reducing the pressure at the regeneration step. On one hand, lower desorption pressure increased the product recovery. On the other hand, the regeneration is more thorough and the bed is cleaner as a consequence. However, the minimal pressure in the industry is limited to a certain value to ensure the economical operation. During the pressurization by high-pressure bed and the feed gas, the bed is pressurized back to the feed pressure and maintained the gas concentration of CH4 at a lower level at the optimal conditions. This benefits from the smaller pressurization valve constants in which step gas-phase concentration increased faster than the solid-phase concentration, as we observed in Figures 3 and 4.

Table 7. Optimal Decision Variables, Their Upper and Lower Bounds, and Optimal Performance Variables for PSA Optimization decision variable

initial value

LB

Process Design 4 1 1.5 0.5 8 2 1.6 0.2 18 2 3 2 2 0.5 2 0.5 2 0.5 Operational 18 15 6.8 2 15 10 Performance Variables 81.44%

CvVW CvVW0 CvVFP CvVRP CvVD CvVP Cv7 Cv8 Cv9 FC1 FC2 FVU purityCH4

UB

optimal value

5 10 10 4 25 8 5 5 5

3.13729 7.57698 2.12586 3.35 23.9476 7.800610 0.609687 0.609687 0.609687

25 10 20

18.7979 6.108018 16.2905 80.00%

recoveryCH4

94.37%

97.88%

power consumption

0.180025

0.1968698

Table 8. Solver and Convergence Statistics for the Optimization of the VPSA System parameter number of axial nodes discretization scheme DAE solver absolute and relative convergence tolerance number of NLP iterations number of NLP line searches NLP solver convergence tolerance CPU time(s) fraction of CPU time spent on sensitivity integration machine details

value 30 CFDM 10−5 29 50 10−4 230290 66.54% Intel Core i5-2100@ 3.10 GHz

4. DESIGN OF THE CONTROL SYSTEM The PSA system is a nonlinear and intrinsically dynamic system in which process variables interact with each other in a very complex manner. It is almost impossible to regulate all of those variables simultaneously. Therefore, determining the manipulative variables which have a potential influence on the process performance (product purity and recovery) and operation safety is of crucial importance during the design of the targeted controller. For the sake of controlling the PSA process, the MPC scheme that requires a detailed process model and a computationally expensive optimization strategy may not be required. A more computationally efficient PID control scheme can be used and employed in this paper because of its simplicity and practicality. It is significant to decide the control variable and manipulative variable initially. For an optimal PSA process aimed at heavy component enrichment, there’s no doubt that the product purity should be the control variable. The choice of the manipulative variable becomes critical for a highly interconnected PSA system. Among all the variables, the time

Figure 2. Comparison of time evolution of the pressure swing over a PSA cycle at the initial and optimal conditions.

significant reduction which contributed significantly to the enhancement of product recovery. When it turned to the end of depressurization equalization step, the adsorption front moved F


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Figure 3. Comparison of axial distribution of concentration of gas CH4 at (a) initial and (b) optimal conditions.

Figure 4. Comparison of axial distribution of concentration of solid CH4 at (a) initial and (b) optimal conditions.

controller, where purity is the control variable and feed step duration is considered as the manipulative variable. It comprises the following steps. First, eliminate the integral and derivative control action after the process has reached steady state and set Kc equal to a small value. At the same time, place the controller in the automatic mode. Afterward, gradually increase Kc in small increments until the ultimate gain and ultimate period is obtained in the purity−feed step duration loop. In the next step, calculate the initial values of the PID controller parameter, namely, the proportional gain, integral time, and derivative time using the Z-N tuning relations. At last, refine the Z-N controller parameter settings by introducing a small set-point change and observe the closed-loop response until the best results in terms of two performance indicators are obtained. The step disturbance procedure described in section 4.1 is employed for this purpose. From the point of real-time control, it is very necessary to add some additional constraints on the maximum and minimum values of feed step duration. On one hand, long feed step duration can over saturate the bed and increase the difficulty of steps such as blowdown and evacuation. What’s more, once the adsorbent is damaged, even longer duration of regeneration steps may not result in the bed achieving a desired clean state. On the other hand, very low value of feed step duration means inefficient utilization of the whole bed and causes the decrease of the unit productivity of the whole PSA system. In addition to the above constraint, large changes in the feed step duration (Δτ) should also be constrained to avoid

duration of the feed step becomes the most promising one because it is closely related with product purity (control variable) and it is able to be regulated in real-time by a programmable logic controller (PLC) in the actual separation process. It is a well-known fact that PSA operation is periodic and the duration of all steps is highly interconnected. To guarantee the sequential control of PSA process and maintain inter-related step conjunction, some assumptions are made toward the duration of all the process steps. It is assumed that the process steps have the time durations shown in eq 18. t ED = t ER = t BD = t PR = t VU1 = t RP1 = 50 s tAD = t VU2 = t RP2 = τ tcycle = 300 s + 3τ

(18)

In the rest of this paper, τ will be referred to feed step duration, namely, manipulative variable (see also the Notation section). The duration of other steps remains constant, and the duration of step AD VU2 and RP2 keeps changing according to the output of the controller; then they in turn change and repeat after every cycle (eq 18) in a closed loop setting. A change in feed step duration not only changes the time duration of adsorption phase (during which the heavy component enters the bed) but also increases the time duration of VU and RP steps, which are directly linked to the product purity and recovery. During one complete cycle, the cycle step settings are uniform for the three adsorption beds. The systematic Ziegler−Nichols (Z-N) continuous cycling method described in Seborg et al.47 is adopted for design of the G


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Industrial & Engineering Chemistry Research over saturation (high Δτ) or avoid sudden surge of inflow (low Δτ).34 These constraints ensure that the controller output purity lies within the upper and lower bounds. The designed control system together with condition constraints are shown below. The corresponding PID tuning parameters are listed in Table 9.

(6) Feed temperature step increase by 10 K and composition step change (methane decrease from 30% to 27%). (7) Feed composition step change (methane decrease from 30% to 27%). After these various disturbances were introduced to the optimization-based PID system, the product purity experienced great change from the desired outcomes in the form of drastic oscillation of the composition, temperature, and pressure during the whole cyclic operation. Here, the high-fidelity PDAEs model acts as the virtual plant. The open loop simulation results of the seven disturbance cases are listed in Table 10. 4.2. Closed Loop Performance. The controller performance is presented in terms of two performance indicators: the controller response time, defined as the minimum number of PSA cycles required to stabilize the methane purity levels around the set point of 80%, and the maximum deviation from product purity generated by the controller. Table 11 provides a summary of closed loop performance with PID controller for those various disturbance scenarios. To better evaluate the control performance, several processes involving the external disturbance have been chosen as typical cases and fully analyzed. Effects of external disturbance on the steady-state process are explained in the follwoing figures. Specific temperature, composition, and pressure responses during the whole cycle to various perturbations are also shown in the following part in addition to the controller performance. 4.3. Control Performance with Single-Variable Disturbance. To analyze the performance of the designed controller, the disturbance scenario 1 in which the feed flow rate experienced a step decrease by 5% was chosen as a typical case with single-variable disturbance. 4.3.1. Control Performance Analysis. The purity limit has been considered to be 1% more or less than the set point. It can be observed in Figure 5 that the methane product purity violated the limits when the feed flow rate experienced a step decrease by 5%. It is highly desired to have a controller which can maintain the purity within the specified control limits irrespective of the disturbances. After the PID controller was introduced, the product gas purity started to go back to its setpoint value gradually in the initial 20 cycles and stabilized at 0.8 with the regulation of feed step duration as described in Figure 6. Meanwhile, the fluctuation extent of product gas purity turned smaller than before, with its improved worst point 79.6% compared with 79% (the worst point of disturbance without control). Therefore, the design of the PID controller with constraints played a very good role in the PSA process with step disturbance, which means that the controller performance is satisfactory. The overall function of the controller, as defined in eq 19, can be divided into three parts: the integrated term (I), proportional term (P), and derivate term (D). The individual control term had different values during the control process. The derivative term was higher initially and dominated the whole control; afterward, the integral and proportional terms increased with time, which is visible in Figure 7, and implied that all the PID terms were in

∂error = 100(purityCH − Ps) 4 ∂t e(k) = 100(purityCH − Ps) 4

PID = Kc(e(k) − e(k − 1)) + Kce(k)

Ts + Kd error τI

τ min ≤ τ ≤ τ max Δτ min ≤ Δτ ≤ Δτ max if |e(k − 1)| > |e(k)| then τ(k) = τ(k − 1) − 0.5PID elseif P > 0 then τ(k) = τ(k − 1) − PID else τ(k) = τ(k − 1) − 10PID (19)

Table 9. Parameter Settings for the PID Controller Ps

Kc

Kd

τI

τmax (s)

τmin (s)

Δτmax (s)

Δτmin (s)

0.8000

0.2

0.001

900

250

150

20

−20

Here, Ps is the set point of product purity, τ the time duration of the feed step in seconds, k the number of PSA cycles, and Ts the sampling interval. The sampling interval cannot be a fixed value for the PSA process because the process steps change and repeat after every cycle, whose duration in turn keeps changing in a closed loop setting. To overcome this challenge, purity−feed step duration measurements are recorded every cycle instead of fixed length of time. Kc, Kd, and τI are the proportional gain, derivative constant, and integral time constant for the PID controller, respectively. 4.1. Open Loop Performance. To make the simulation results of the PSA process closer to engineering practice, seven different types of external disturbance in the feed condition were designed and are listed below. Disturbance scenario: (1) Feed flow rate step decrease by 5%. (2) Feed flow rate step decrease by 5% and temperature step increase by 10 K. (3) Simultaneous step disturbance of feed flow rate (by 5%) and temperature (by 10 K) and composition (methane decrease from 30% to 27%). (4) Feed flow rate step decrease by 5% and composition step change (methane decrease from 30% to 27%). (5) Feed temperature step increase by 10 K.

Table 10. Open Loop Simulation Results for Seven Disturbance Cases disturbance scenario

1

2

3

4

5

6

7

set point of purityCH4 (%)

80.00

80.00

80.00

80.00

80.00

80.00

80.00

purityCH4 (%)

78.79

77.92

76.14

76.99

79.17

77.47

77.42

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Industrial & Engineering Chemistry Research Table 11. Closed Loop Simulation Performance for Seven Disturbance Scenarios disturbance scenario

1

2

3

4

5

6

7

respond time (cycles) maximum deviation of product purity (%)

18 79.56

18 79.11

16 78.47

15 78.64

19 79.62

14 79.14

14 79.15

Figure 5. Regulation of manipulated variable and corresponding change of control variable (purityCH4) after the introduction of constrained PID controller (disturbance scenario 1).

Figure 7. Regulation of individual control items of the constrained PID controller.

see that even though the feed flow rate had decreased, purity had quickly returned to its desired set point, 80%, with the regulation of feed step duration. In the pressurization and adsorption step, the amount of methane that adsorbed on the adsorbent decreased. This was consistent with the temperature revolution in Figure 8. When it turned to the depressurization equalization step, the gas concentration of light component nitrogen was so high that its exclusion amounts from the outlet in the same duration became larger, which was a slight advantage over the original process. Nevertheless, the much lower purity of methane in the replacement gas had deteriorated effect of replacement step, as Figure 9 showed. As a consequence, the product gas came from blowdown and evacuation step became degraded significantly, as observed from Figure 10. In the face of the above troublesome dilemma, the designed constrained PID controller had proved itself versatile and powerful through precise adjustment of feed step duration. The temperature and composition of the whole PSA process had returned to the initial steady state with the efficient control of the PID controller. 4.4. Control Performance of Multivariable Disturbance. Single-variable disturbance such as feed flow rate and feed temperature can represent the real operation of chemical engineering units to only some extent. The random combination of those single variables will indicate the real conditions in a more complex way and provide meaningful theoretical foundation for the control of chemical processes. The regulation results of constrained PID controller in these complicated conditions will demonstrate its practicability and effectivity. Analyzing all those disturbance scenarios is such a huge amount of work that choosing the following two typical scenarios will be the wise chocie: scenario 2, feed flow rate step decrease by 5% and temperature step increase by 10 K; or scenario 3, simultaneous step disturbance of feed flow rate (by 5%) and temperature (by 10 K) and composition (methane decrease from 30% to 27%).

Figure 6. Comparison of purityCH4 in two cases: (1) disturbance without control and (2) embedding the constrained PID controller after disturbance (disturbance scenario 1).

action. After about 20 cycles, the three items became stabilized at zero. 4.3.2. Process Analysis. To analyze the controller performance thoroughly, the axial distribution of bed temperature and gas-phase CH4 after several process steps have been plotted and compared. As we can see from Figures 8−10, the feed flow rate disturbance did induce a visible change in the temperature and concentration at the outlet side of the bed. The effect of step disturbance was more pronounced in the first few cycles. Because the product purity and recovery are mainly dependent on the replacement and evacuation step, any changes made to those steps can affect the process performance. It is represented in Figures 8−10 that there was still much light component in the bed at the replacement step after disturbance. As a result, the product got from the replacement and evacuation step degraded. After the introduction of the PID controller, we can I


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Figure 8. (a) Bed temperature and (b) gas-phase CH4 concentration profile after the adsorption (AD) step in three kinds of scenarios: (1) initial steady-state condition, (2) disturbance without control, and (3) control after disturbance (disturbance scenario 1).

Figure 9. (a) Bed temperature and (b) gas-phase CH4 concentration profile after the replacement (RP2) step in three kinds of scenarios: (1) initial steady-state condition, (2) disturbance without control, and (3) control after disturbance (disturbance scenario 1).

Figure 10. Bed temperature and gas-phase CH4 concentration profile after the evacuation (VU2) step in three kinds of scenarios: (1) initial steadystate condition, (2) disturbance without control, and (3) control after disturbance (disturbance scenario 1).

product purity declined rapidly in the first 10 cycles and severely violated the control limits in the open-loop scenario. A well-behaved controller should stabilize the control variable to the desired point at a reasonable speed without drastic changes. Larger change in the feed step duration in both scenarios was observed in Figure 11 to offset the negative influence the disturbance brought. From Figure 11, we could see that even though the system needed the largest feed step duration, the product purity had returned back to the set point. The tendency of corresponding PID terms was almost the same but with larger amplitude. The PID terms had also experienced the largest variation in scenario 3. In the first cycles, the integral

5. CONTROL PERFORMANCE ANALYSIS Operation conditions and parameter settings of scenario 2 were the same as those of scenario 1 except for the extra disturbance of feed temperature. Scenario 3 had considered the simultaneous step disturbance of feed flow rate and temperature and composition. In Figure 12, which is a direct revelation of the influence of the two disturbance scenarios, we could see that product purity had dropped below 78% quickly in no more than 15 cycles at the presence of disturbance scenario 2 which was very undesirable during actual plant operations. In the worst scenario, in which there is simultaneous step disturbance of feed flow rate, temperature, and composition, the methane J


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Figure 11. Regulation process of feed step duration and corresponding change of purityCH4 after the introduction of constrained PID controller. Left panel, disturbance scenario 2; right panel, disturbance scenario 3.

Figure 12. Comparison of purityCH4 in two cases: (1) disturbance without control and (2) embedding the constrained PID controller after disturbance. Left panel, disturbance scenario 2; right panel, disturbance scenario 3.

Figure 13. Regulation of individual control items of the constrained PID controller. Left panel, disturbance scenario 2; right panel, disturbance scenario 3.

K


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Hb = bed height (m) Kc = proportional gain for PID controller Kd = differential constant for PID controller kLDF,i = LDF rate constant of component i (1/s) kg = effective axial thermal conductivity (W/(m·K)) M = molecular weight of the gas (kg/mol) P = pressure (Pa) PID = value of the PID controller Ps = set point of purity purityi = purity of component i purityCH4 = control variable Qfeed = flow rate of feed gas (m3/s) qi = adsorbed phase concentration of component i (mol/kg) qi* = adsorbed phase concentration in equilibrium with bulk gas of component i (mol/kg) qm,i = maximum adsorbed phase concentration in equilibrium with bulk gas of component i (mol/kg) R = ideal gas constant (J/(mol·K)) Rb = bed radius (m) RP = particle radius (m) recoveryi = recovery of component i τI = integral time constant for PID controller t = time (s) T = temperature (K) τ = manipulative variable (s) τmax = upper limit of the time duration of the feed step (s) τmin = lower limit of the time duration of the feed step (s) Δτmax = upper limit of the change of time duration of the feed step (s) Δτmin = low limit of the change of time duration of the feed step (s) Ts = sampling interval Tw = wall temperature (K) vg = superficial velocity (m/s) wads = mass of adsorbent (kg) yi = molar fraction of component i z = partition of the bed length Hb (m)

and proportional terms were the dominating terms, then the derivative term became major one. After about 25 cycles, the derivative item became larger again to stabilize the purity to the set point, which is visible in Figure 13. As displayed in Figures 11 and 12, the worst scenario with the simultaneous change of feed flow rate, temperature, and composition experienced satisfactory control performance. This also demonstrated the practicability and feasibility of the constrained PID controller to maintain the product quality at various kinds of disturbances.

6. CONCLUSIONS This work presents a systematic framework for the simulation, optimization, and control of a PSA system while utilizing a detailed mathematical model. Important PSA objectives, decision variables, and operational constraints are discussed and incorporated in the optimization framework. The best closed loop recovery obtained for the PSA system under consideration is around 98%, with the additional purity and energy consumption requirements. Embedding controller stability and robustness (in the form of additional constraints) in the controller synthesis formulation, though it is expected to complicate the overall design methodology, have already been investigated. Seven disturbance scenarios with different complexity have been introduced to the optimization-based PSA process. The closed loop performances with the constrained PID controller have also been compared with open loop simulations. The control system has proved its ability to reject various unknown disturbances and gives a satisfactory performance in all scenarios. Depending on a particular operational configuration, the control system can be adapted by the addition of the required constraints and further used for the control of the PSA system.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +086-022-27892097. E-mail: [email protected]. Notes

Greek Letters

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Program of National Fund 51,178,300. NOTATION

Roman Letters

■

b0 = adsorption affinity (1/KPa) Cpg = constant pressure specific heat of the gas mixture (kJ/ (kg·K)) Cps = specific heat of the adsorbent (kJ/(kg·K)) Cv = valve constant (mol/(KPa·s)) ci = gas phase concentration of component i (mol/m3) Dax = axial dispersion coefficient (m2/s) Dci = effective diffusion coefficient of component i (m2/s) Dk,i = Knudsen diffusion coefficient of component i (m2/s) Dm = molecular diffusion coefficient (m2/s) dp = particle diameter (m) Dv = molecular diffusion volume (cm3/mol) e = bias of purity error = integral bias of purity F = volume flow of rates (mol/s) h = heat transfer coefficient between gas and column wall (W/(m2·K))

εb = bulk phase porosity εp = particle phase porosity ρp = density of the adsorbent (kg/m3) ρg = density of the gas phase (kg/m3) ηp = work efficiency of compressor and vacuum pump τ = pore tortuosity γ = compressor adiabatic factor ΔHi = isosteric heat of adsorption of component i (kJ/mol)

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A Systematic Simulation and Proposed Optimization of the Pressure

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