Ind. Eng. Chem. Res. 2003, 42, 5287-5292
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A Practical Method for the Dynamic Determination of the Product Oxygen Concentration in Pressure-Swing Adsorption Systems Chris C. K. Beh and Paul A. Webley* Department of Chemical Engineering, Monash University, P.O. Box 36, Victoria 3800, Australia
Pressure- and vacuum-swing adsorption processes are challenging to model and difficult to solve rapidly because the system response is a nonlinear function of both the axial and temporal domains with periodic boundary conditions. Extensive computing power is required to solve the conservation equations that describe the temperature, composition, and pressure profiles during operation of an adsorption process. Furthermore, the physics of the source/sink terms in the conservation equations of mass and energy, which relates to the ad/desorption of absorbable species, is not always easily described. These rigorous numerical models are useful for furthering our understanding of these complex processes and are the only methods available for the design of industrial units. However, such complicated numerical simulators for model-based control schemes are not feasible at the current level of computing resources. Simplification of the conservation equations is required to derive a practical mechanistic model for predictive control purposes. In a previous study, Beh and Webley (Adsorpt. Sci. Technol. 2003, in press) have demonstrated that much of the complexity of these processes can be captured through the use of a simple model consisting of a series of coupled tanks which approximates the bulk flows and pressures to a satisfactory degree. In this paper, an extension is made to this model to incorporate the time-varying composition variable. The limitations of this method will be discussed in relation to field operation. Introduction Oxygen enrichment technologies such as vacuum- and pressure-swing adsorption (VSA and PSA, respectively) are economic alternatives to high-purity oxygen produced by cryogenic distillation and are increasingly being used to improve productivity and lower the running costs of the current processing plants.1,2 Hence, understanding this process for the manufacture of this important chemical commodity has gained increased academic interest over the past 20 years.3 Unlike many common chemical engineering unit operations, PSA and VSA processes operate batchwise under non-steadystate, nonequilibrium conditions. The level of complexity is increased in the air enrichment system in which bulk gas separation occurs, causing nonuniformity of the axial velocity and thermal gradients in the adsorber column. Some methods exist for the analysis and preliminary design phase of these systems,4 but their complexity has forced the use of detailed mechanistic models whose solution procedure requires extensive computing power. Although there has been some development in solution acceleration methods for these systems (as discussed in the reference by Wilson and Webley5), the use of these mechanistic models as the basis of a model-based controller structure or process monitoring is far from practical. Therefore, simplification of the conservation equations is required to derive a mechanistic model that not only captures the underlying physics but also is simple enough to be utilized in real time. Our primary motivation in developing a simple mechanistic model is to significantly reduce computation time while still providing sufficient ac* To whom correspondence should be addressed. Tel.: +61 3 9905 1874. Fax: +61 3 9905 5686. E-mail: paul.webley@eng. monash.edu.au.
curacy to permit use of the model in real time for either process modeling or control. In a previous study, Beh and Webley6 have demonstrated that much of the complexity of the PSA process can be captured through the use of a simple model consisting of a series of coupled tanks, which approximate the bulk flows and pressures to a satisfactory degree. The objective of this paper is to extend this existing model to permit the addition of the time-variant composition variable, which is arguably the most important variable from a customer’s point of view. The paper will conclude by discussing its performance and limitations in comparison with field experiments. Development of the Oxygen Purity Model A simplified dynamic model of the VSA processes called SoCAT (simulator of coupled adsorptive tanks) was developed in previous work.6 This numerical model considered an isothermal system whereby the two adsorber beds were modeled as two lumped systems with no spatial dependencies. The adsorbers were linked to a feed and product tank, and the model was solved in a cyclic manner using the boundary conditions of the pilot-plant cycle. The intended goal of this model was to correctly predict trends in short-term responses and facilitate understanding of the pressure-flow interactions. The model was not intended to replace existing detailed simulators that are used to facilitate the indepth understanding of physical features such as mass transfer. We found that this simple model described the bulk flows and pressures well when compared to data gathered from pilot-plant studies. That model, however, was not capable of predicting oxygen product purity, which is one of the most important variables for control. Because the oxygen concentration is dependent on spatial information and our model was spatially lumped,
10.1021/ie021060b CCC: $25.00 © 2003 American Chemical Society Published on Web 09/13/2003
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it was intrinsically unable to predict oxygen purity. We demonstrate below how dynamic prediction of oxygen purity can be made by extending our simple model without adding great complexity or significant computing requirements. The model derivation begins by performing a component mass balance around the product tank, yielding
d(Npyp)/dt ) niyi - noyo
(1)
where Npyp are the moles of oxygen in the product tank. The new unknown variables in eq 1 are mole fractions yi, yp, and yo. In the case where perfect mixing is assumed in the product tank, the mole fraction of oxygen (yo) in the stream leaving the tank is equal to the mole fraction in the tank (yp). The total moles in the product tank, Np, and the mole flow rates into and out of the tank, ni and no, respectively, are obtained by integration of the total mass balances around the feed, product, and adsorber vessels and are performed in SoCAT. Equation 1 is therefore decoupled (i.e., solution of the system mass balances is not influenced by yp) from the equations describing bulk flows in the system and can be solved independently. To solve eq 1, knowledge of the inlet oxygen mole fraction, yi, is required. To estimate the inlet mole fraction, yi, it was assumed that the actual response of the adsorber beds to perturbations is “instantaneous” (i.e., movement of the adsorbate front occurs within the same cycle that the disturbance is made), but this response is dampened by the capacity and the flows into and out of the product tank. This assumption is valid because control action is taken on a cycle-based time scale, not on an instantaneous time scale. When a perturbation to the process occurs, there is a change to the number of moles produced by the bed undergoing feed. This change in the number of moles is accompanied by an increase or decrease in purity according to whether the adsorption front moved through the end of the bed or was retained within the bed. Thus, the “pulse” of the product gas produced by the bed will have a new purity (and quantity) produced on the same cycle that the perturbation was made. The relationship between the average purity and the number of moles produced from the bed is therefore the same dynamically and at cyclic steady state (CSS). Of course, the product vessel from which the customer supply of oxygen is ultimately taken contains “pulses” of product gas of varying purity from several previous cycles and so does not reach CSS within the same cycle as the perturbation. It is this lag between the “instantaneous” response of the bed and the accumulated history of previous cycles “stored” in the product tank that eq 1 attempts to represent. The main reason product vessels are employed in an industrial situation is to dampen the pressure, flow, and product concentration swing. Typically, these large installations can have product tanks sized between 4 and 5 times the volume of the adsorber beds, and it is this large capacity that permits the suppression of fluctuations. Figure 1 shows the cyclic steady-state oxygen purity as a function of the number of moles of product produced with a second-order polynomial fit through the data points. It shows reasonable correlation (R2 ) 0.79), although the difference between the model and the experimental values can be up to (2% in some cases, which may cause some errors in the model approximation. This simple empirical model was extrapolated over the range of product flows with bounds placed on the
Figure 1. CSS relationship between product flow per cycle and oxygen purity. Comparison of the empirical fit to pilot-plant data; R2 ) 0.79.
empirical fit such that compositions less than and greater than that physically attainable are not permitted (i.e., 21% e yi e 95%). It should be noted that the maximum obtainable oxygen purity using PSA is 95% because argon shares equilibrium adsorption properties similar to those of oxygen. Typically, the product concentration is measured downstream of the product tank because this is the only value of interest to the customer. However, the oxygen inlet mole fraction, yi, that is obtained from Figure 1 is an averaged value, and hence the instantaneous mole rate and composition are estimated by
ni ≈ n j i ) Ni/τc
(2)
yi ≈ yji
(3)
where τc is the total cycle time. To solve eq 1 correctly, a relationship for niyi is required, which we approximate by the averaged value as described in eq 4.
niyi ≈ n j iyji ) Niyji/τc
(4)
In reality, the gas entering the product tank is not fed continuously but enters as a nonuniform pulse of fixed period due to the ad/desorption steps required in VSA/PSA processes. Equation 4, therefore, is better represented by eqs 5 and 6. These equations account for the steps in which product gas from the adsorbers is either being fed into the product vessel (eq 5; therefore, the average molar flow of the component of interest is calculated) or not being fed (eq 6; hence, the molar flow for the species is zero)
n j iyji ) Niyji/τp ) C n j iyji ) 0 ) C
t ) tp t * tp
(5) (6)
where C is a constant based on eqs 5 and 6, τp is the product feed time, and tp is the instantaneous time corresponding to the production step in the cycle. Therefore, over a cycle the assumption of an average niyi holds as
∫0τ niyi dt ) ∫0τ C dt ) Niyji c
c
(7)
It should be noted that the instantaneous mole fraction, yi, entering the product vessel within the cycle does not correspond to actual/measured values due to the as-
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5289 Table 1. Summary of the Pilot-Plant and Oxygen VSA Cycle Studied Pilot-Plant Details no. of beds 2 total bed height (mm) 1810 internal bed diameter (mm) and 110 × 3.0 wall thickness (mm) bed material PVC insulated with fiber glass wool with reflective aluminum outer adsorbent lithium zeolitic sieve, 9.1 kg/bed prelayer length (mm) 300 prelayer alumina, 2.5 mm beads, 0.5 kg/bed particle diameter dp (mm) 1.4 feed stream composition ∼22% O2, ∼1% Ar, ∼77% N2; (% molar) dew point -50 °C at atmospheric conditions mass balance closure at CSS -3% to -1%
sumption of averaged mole fraction as given by eq 8.
yi ) f[ni(t)] * yji ) g[n j i(τc)]
(8)
From eqs 5 and 6, eq 1 can be rewritten as
Vp d(Ppyp) ) C - noyo RT dt
(9)
If we assume that the tank is well-mixed, then yo ) yp and therefore eq 9 can be rearranged to yield
dPp RT (C - noyp) - yp Vp dt dyp ) dt Pp
(10)
Equation 10, with application of the assumption for yi (determined empirically from Figure 1), provides a simple relationship for yp, which is entirely decoupled from the total mass balance of the adsorber-tank system (to reiterate, the mass balance equations are not dependent on the solution of yp). A final note with regards to the implementation of this method is that the once the total mass balances are solved to determine the molar parameters in eq 1 (i.e., integrate for No), the time integrator must be reset to the initial conditions at the commencement of the cycle. The solution vector from the ordinary differential equation solver now includes eq 10. Results and Discussion The experimental data presented in this paper are based on the VSA cycle and pilot plant described in detail in a previous publication.6 To aid brevity, a summary specification of the pilot plant is listed in Table 1 with the cycle details shown in Figure 2. Although the system response to several step perturbations was studied, in this paper, only the response to purge valve perturbations will be discussed in detail, although the findings for the other valve responses are summarized in Table 2. The purge valve was chosen as an example for the application of this method because the effect of the purge valve is the most difficult to comprehend and hence model. Its effect on the oxygen concentration profile is twofold. 1. The purge valve varies the product concentration leaving the adsorber system by shifting the position of
Figure 2. Sequence diagram for a dual-bed, six-step O2 VSA cycle. Total cycle time ) 60 s. Table 2. Phase Angle and Dead-Time Estimation of the Pilot-Plant Product Stream Averaged Oxygen Concentration to Step and Frequency Perturbations valve feed
purge
product
period (cycles) step response 5 10 15 step response 5 10 15 step response 5 10 15
phase angle (deg) -297 -216 -160 -223 ∼-122 -107 -298 -219 -188
τD (cycles) 2 3 4 4 1 2 1 > τD < 2 2 3 3 4 5
the mass-transfer zone axially (this may also cause some degree of dispersion of the adsorption front). 2. The purge valve alters the composition profile leaving each individual bed during the production step (steps 2 and 5 in Figure 2) as purge gas of varying purity (typically high to low impurity) enters the bed receiving purge (as shown in steps 3 and 6 of Figure 2). In addition to step perturbations, frequency perturbations were made. In this case, the purge valve position was updated sinusoidally every cycle at the commencement of step 3 (refer to Figure 2). Figures 3 and 4 show the performance of our simple model in approximating the purity response to a step and periodic change in the purge valve in comparison with measured data. The
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Figure 3. Response of the cycle-averaged oxygen purity to a 3% step change in the purge valve position. The triangle data points show the model output delayed by a single dead time. Step change initiated at cycle 5. Pilot-plant steady-state gain ∼ 1.1%. Model steady-state gain ∼ 1.05%.
Figure 4. Response of the cycle-averaged oxygen purity to a cyclic purge valve input of amplitude of 3% and a period of 10 cycles. The triangle data points show the model output delayed by one dead time. Frequency perturbation initiated at cycle 5. Pilot-plant amplitude ∼ 0.39%. Model amplitude ∼ 0.63%.
model yields reasonable results with modest differences, capturing the effects of the step change (within 0.3%; Figure 3) well and the amplitude in the frequency response scenario (0.63% for the model compared to 0.39% measured by the pilot plant; Figure 4). These differences in the estimation of the absolute value of the composition variable are sufficiently small such that the application of this technique in the field for process monitoring or control would yield useable results because the oxygen purity is commonly permitted to drift within 1-2% of the set point. This process/model mismatch is in part due to the accuracy of the initial calibration of the SoCAT model. For instance, a previous study has shown that this discrepancy can be up to 20 mmol/cycle between experimental data and the model.6 This variance amounts to approximately an oxygen purity difference of 0.6% in the absolute value between plant and model. In addition to the problems associated with the initial model calibration, incorrect estimations of the change in magnitude of the change in boundary conditions (e.g., valve flow coefficients or product backpressure) account, in part, for some of the mismatch. Beh and Webley6 have indicated that a difference of up to 33 mmol/cycle for a step change between the change in product flow predicted by the model in comparison with the plant was observed. By assuming a linear relationship over time (i.e., on a per cycle basis), 33 mmol/cycle corresponds to
a 1.0% difference in the predicted purity. For the purge valve step response case, this difference is marginal (∼4 mmol/cycle), and hence there is less mismatch between the SoCAT model and the plant (refer to Figure 3). Another cause of discrepancy between the model and plant data is the reliability of the experimental data, which show some scatter of approximately 2% at the same product flow (refer to Figure 1). Consequently, the empirical fit obtained may result in similar differences when compared to the pilot plant. What is encouraging though, as is signified in Figures 3 and 4, is the ability of the model to correctly predict the transient responses. It is cautioned, however, that for faster cycles such as rapid PSA the assumption of instantaneous product concentration leaving the adsorber vessels may no longer hold. The principal error between experimental data and the model output is due to the poor mixing characteristics that occur within the product tank. For this pilot plant, nonideal mixing manifests itself as a time delay and thus the model output is shifted by an appropriate amount to account for this. In the case of the oxygen VSA pilot plant, flow into the 60-L product tank was fed by 1/2-in. tubing running perpendicular to the vessel wall located near the top of the tank, while the exit stream was located near the bottom of the vessel and also runs perpendicular via a 1/2-in. tube. No baffling or internal mixing was used in the tank. On average, for the set of experiments conducted, approximately 20 L of product gas entered and exited the tank at CSS. If the conditions were dominated by pure plug flow, then a step response in the exit product stream would result in a stepwise change after a delayed period (dead time of around three cycles in this case ≡ volume of product vessel/volume of gas exiting the vessel per cycle). On the other hand, if conditions within the product vessel were well-mixed, then the measured exit composition would follow an asymptotic relationship, with the time constant of the response being a function of the capacity of the tank. However, as the results shows, a condition somewhere between plug flow and well-mixed exists and the system dead time, which is a function of both the volume of flow into and out of the product tank and of the flow profiles within the tank itself, is not constant, as presented in Table 2. It reveals that the time delay, τD, varies between step tests and valve cycling and also varies with the type of disturbance. The phase angle, which is a measure of the lag between the input and output response, decreases with increasing period as lower frequencies exert incrementally smaller changes to the system, thus allowing it more time to react. For completeness, the full response time (time to reach 100% of the value) of the paramagnetic analyzer and connecting tubing was measured and found to be about 25-30 s for a step change in composition at constant flow. Because the response of the analyzer is much less than the period of the cycle studied (the total cycle time is 60 s), this further reinforces the hypothesis that the observed delay is attributed to poor mixing in the tank. In process engineering, tracer experiments are often used to characterize the transport capability of the reactor vessels usually assuming steady-state relationships.7 However, these general methods are difficult to apply to the unsteady-state oxygen VSA process because it is these transient, instantaneous flows that alter the composition profile in the product vessel. Therefore, to estimate the extent of mixing or to establish models for
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mixing, the time-variant pressure profile of the tank and the flows in (which are a series of nonuniform pulses) and out would have to be physically simulated while ensuring that the tracer is nonintrusively introduced. Various permutations of the flow and pressure regimes would have to be tested to eventually obtain a dynamic model of mixing. Work on the development of simple mixing models for PSA/VSA systems might be useful for the initial “grass roots” design and cost of industrial installations. This inability to characterize the mixing profile in the product tank can lead to severe controllability problems. The typical industrial convention is to use multiple proportional-integral-derivative (PID) loops to control variables such as purity, flow, and pressure.8 Once the controller pairings have been made and decouplers are applied, controller tuning becomes the principal control problem. In the case of PID controllers, there are two main tuning methodologies: open-loop step responses and frequency perturbations (open or closed loop).9 For each of the methods mentioned, there exist numerous algorithms in which to establish estimates of the controller settings.9 However, most of these methods assume that the system can be identified by an empirical fit through an “experimental” data set (which can be gathered from either the field or model data) or by direct synthesis of the model and controller to obtain appropriate gains.10 If model parameters such as the system lag, gains, or time delays varied with time (due to material deterioration or fouling for example), then the calculated controller gains would no longer be valid for the system. This may result in major robustness problems and controllability issues. Various methods have been developed in the past that involve the autotuning of controller settings or gain scheduling to help overcome some of these difficulties.9 However, as shown above in the case of oxygen VSA, it makes no sense to incorporate a gain schedule, for example, that updates the controller settings as a function of the flow and disturbance regime. Other issues such as the controller “kick” arise, which may result in deficiencies in control. Contrary to that previously discussed, the theoretical development of controller tuning rules for batch processes such as oxygen VSA still remains in its infancy. For example, the appropriate time variable for a batch process is the cycle number, and consequently it is possible that a particular controlled variable has no time delay (such as product flow per cycle as discussed by Beh and Webley6). If the product vessel were ideal, then the plant would behave as the simple model predicts and hence an instantaneous response would result. This situation has not been adequately resolved in the current literature with regards to PID tuning, which always assumes a time delay, with the controller settings calculated as a function of this delay.9 In the case of oxygen enrichment, the primary variable of importance to the customer is the product concentration, with the majority of commodity gas users being large industrial installations processing up to 5-80 metric tonnes of contained oxygen per day at 9395% purity.1 Hence, product vessel capacities range from several hundred to thousands of cubic meters for these large users. In these cases, the inlet and outlet flow arrangements and possibly the internals should be designed such that uniform composition occurs throughout the tank and concentration gradients are avoided.
It is envisaged that the additional outlay in cost spent designing well-mixed product vessels would quickly pay itself back with reduced rework, fewer control problems, and a generally satisfied customer. Conclusions We have derived and demonstrated the application of a simple empirical model that represents the complicated dynamic composition profile of an oxygen VSA process to a usable degree. This finding suggests that the physics of the processes has been correctly assumed and emphasizes the usefulness of this technique as the basis for a model-based control structure and for overall processes monitoring. The sources of error in this method involve the calibration of the valve flow coefficients and the availability of reliable experimental data in which to establish a relationship between total product flow and product concentration. The primary cause of mismatch between the process and model has been identified to be the nonidealities that exist in the pilot-plant product tank, which tends to shift the output profile in time rather than alter its absolute value. It is envisaged, therefore, that by improved design of these vessels benefits would be gained from the decreased downtime attributed to a reduction in process control problems and subsequently resulting in greater customer satisfaction. Acknowledgment The authors thank Air Products and Chemicals Inc. for their support and funding of this project. Nomenclature C ) constant defined by eqs 5 and 6 n ) mole rate, mol/s N ) moles of gas, mol P ) pressure, kPa‚A R ) universal gas constant, kJ/mol/K t ) time variable, s T ) temperature, K V ) volume, m3 y ) mole fractions of nitrogen and oxygen τ ) integrated time variable Model Subscripts c ) cycle i ) inlet stream o ) outlet stream p ) product tank and product step
Literature Cited (1) Parkinson, G.; D’Aquino, R.; Ondrey, G. O2 Breathes New Life into Processes. Chem. Eng. 1999, Sept, 28-31. (2) Shelley, S. Out of Thin Air. Chem. Eng. 1991, Jun, 30-39. (3) Ray, M. S. Adsorptive and Membrane-type Separations: A Bibliographical Update 1999. Adsorpt. Sci. Technol. 2000, 18, 439468. (4) Knaebel, K. S. The Basics of Adsorber Design. Chem. Eng. 1999, Apr, 92-103. (5) Wilson, S. J.; Webley, P. A. Perturbation Techniques for Accelerated Convergence of Cyclic Steady State (CSS) in Oxygen VSA Simulations. Chem. Eng. Sci. 2002, 57, 4145-4159. (6) Beh, C. C. K.; Webley, P. A. The Dynamics and Characteristics of an Oxygen Vacuum Swing Adsorption Process to Step Perturbations. Part 1. Open Loop Responses. Adsorpt. Sci. Technol. 2003, 21, 319-347.
5292 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 (7) Fogler, H. S. Elements of Chemical Reaction Engineering, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1992; pp 759-795. (8) Beh, C.; Wilson, S.; Webley, P.; He, J. The Control of the Vacuum Swing Adsorption Process for Air Separation. In Proceedings of the Second Pacific Basin Conference on Adsorption Science and Technology; Do, D., Ed.; World Scientific: Singapore, 2000; pp 663-667. (9) A° stro¨m, K.; Ha¨gglund, T. PID Controllers: Theory, Design, and Tuning, 2nd ed.; ISA: Research Triangle Park, NC, 1995.
(10) Chen, D.; Seborg, D. E. PI/PID Controller Design Based on Direct Synthesis and Disturbance Rejection. Ind. Eng. Chem. Res. 2002, 41, 4807-4822.
Received for review December 30, 2002 Revised manuscript received July 10, 2003 Accepted July 11, 2003 IE021060B
