Multi-Objective Optimization of Pressure Swing ... - ACS Publications

Ind. Eng. Chem. Res. 2007, 46, 3751-3765

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Multi-Objective Optimization of Pressure Swing Adsorbers for Air Separation B. Sankararao and Santosh K. Gupta* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

Multi-objective optimization of the operation of a two-fixed-bed, four-stage pressure swing adsorber (PSA) unit for the separation of air is studied using the newly developed multi-objective optimization technique, the modified MOSA-aJG. The rigorous model developed for a single fixed-bed adsorption column is applicable to PSA units provided different and appropriate boundary and initial conditions are used for the four stages, namely, pressurization, adsorption, blowdown, and purge. Seven of the nine model parameters are the same as obtained earlier for a single-bed O2-N2-Zeolite 5A unit. The remaining two parameters are tuned using one set of experimental data on a PSA unit with the same system. Reasonably good agreement is observed between simulation and experimental results. Two two-objective optimization problems and a four-objective problem are formulated for the operation of a PSA. These are solved to obtain optimal values of the adsorption pressure, desorption pressure, adsorption and purge times, and input flow rates during the adsorption and purge stages. The effects of different decision variables on the several objective functions are discussed in detail. Introduction Separation of gases accounts for a major fraction of the production cost in chemical, petrochemical, and related industries. There has been a growing demand for economical and energy efficient gas separation processes. The new generation of synthetic and more selective adsorbents developed in recent years has enabled adsorption-based technologies to compete successfully with traditional gas separation techniques, such as cryogenic distillation. The rapid commercialization of adsorption-based separations, such as pressure swing adsorption (PSA), has drawn attention to the need for better simulation, design, and optimization of these processes. The industrial applications4-6 of PSA include hydrogen and carbon dioxide recovery from reformer off-gases, air separation for producing oxygen- and nitrogen-rich mixtures, gas drying, recovery of methane from landfill gases, and the separation of carbon dioxide and carbon monoxide from blast furnace flue gas and other waste gases from the steel industry. The popular models available in the literature for adsorption in packed beds are the equilibrium,7-11 linear driving force12-15 (LDF), micro-pore diffusion,16-18 macro-pore diffusion,19,20 and bi-disperse pore diffusion21 models. Several workers have considered pressure (velocity) variations,19,22-24 temperature variations,19,21,22,24 and axial dispersion,20,23,25,26 along with one of the basic models described above. However, there is a need for improved models by considering all the effects simultaneously. In this work, a rigorous model,3,27 developed for a single fixed-bed adsorption column, is used for obtaining solutions for the different stages (pressurization, adsorption, blowdown, and purge) of a PSA unit, using the boundary and initial conditions applicable to each of the four stages. A computer code has been developed to integrate the set of coupled partial differential equations (PDEs) for the four stages sequentially. These provide the exit concentrations of the various components of the feed gas in the two beds in each of the four stages, when the system is operating under a cyclic steady state (CSS). This code is used to optimize a PSA unit (for the production of O2) involving the system, O2-N2 on Zeolite 5A. * To whom correspondence should be addressed. E-mail: skgupta@ iitk.ac.in. Tel.: 91-512-259 7031; 7127. Fax: 91-512-259 0104 (attn. Dr. S. K. Gupta).

Until about 1980, almost all systems in chemical engineering were optimized using a single objective function. Often, the objective function accounted for the economic efficiency only, which is a scalar quantity. In most real-life systems, for example, the PSA system, several objective functions can be identified, that are often conflicting and non-commensurate. Often, these objective functions cannot be accommodated meaningfully in a single objective function (e.g., cost or profit) and so one needs to carry out multi-objective optimization. Multi-objective optimization involves the simultaneous optimization of more than one objective function. Several multi-objective optimizations (with constraints) have been reported for industrial systems over the last two decades, using a variety of algorithms. In such cases, one may get a set of several equally good (non-dominating) solutions, referred to as a Pareto28 set/front. These can provide useful insights to a decision-maker, who can then use his or her judgment or intuition to decide upon the “preferred” solution (operating point). Two popular optimization techniques for single objective functions are the simple genetic algorithm29-31 (SGA) and the simple simulated annealing31,32 (SSA). Several extensions of SGA and SSA have been developed to solve optimization problems involving multiple objectives. The vector evaluated genetic algorithm33 (VEGA), the non-dominated sorting genetic algorithm34 (NSGA), the elitist NSGA-II,35 and two jumping-gene (JG) adaptations of NSGA-II, namely, NSGA-II-JG36 and NSGA-IIaJG,37-39 are popular extensions of SGA. Similarly, multi-objective simulated annealing40 (MOSA) and its two JG adaptations, MOSA-JG1,2 and MOSA-aJG,1,2 are popular extensions of SSA. These optimization algorithms are being used extensively for solving multi-objective optimization problems in chemical engineering. The applications of NSGA (and other techniques) in chemical engineering have been reviewed by Bhaskar et al.41 while those of NSGA-II have been reviewed by Nandasana et al.,42 the latter for problems in chemical reaction engineering. A few optimization studies43-46 on PSA systems have been reported in the literature using a single objective function. Cruz et al.43 studied the optimization of small and large scale units of cyclic adsorption processes [PSA and vacuum swing adsorption (VSA)] using the classical Skarstrom47 cycle. They considered profit as the objective function. Choi et al.44 studied

10.1021/ie0615180 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/19/2007

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Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007

Figure 1. Basic two-bed PSA system at any stage of a cycle.

the optimization of PSA by considering the recovery of CO2 as the objective function to find the optimal reflux ratio and the adsorption stage time. Ding et al.45 optimized the PSA system and the temperature swing adsorption (TSA) system by maximizing the product rate in PSA and minimizing the energy input in TSA. Ko and Moon46 studied the optimization of the startup operating conditions of a rapid pressure swing adsorption (RPSA) process. The objective function defined in their study was to reduce the operating power and shorten the time to reach the CSS, as well as to increase the purity of the desired product at the CSS. To the best of our knowledge, no study has been reported in the open literature on the multi-objective optimization of PSA units. In this study, a few multi-objective optimization problems are formulated for a PSA system and solved using the recent and efficient algorithm, the aJG adaptation of the multi-objective simulated annealing (MOSA-aJG). Formulation Process Description. The separation of components of gaseous mixtures in PSA is achieved by the preferential removal of one (or more) species by the adsorbent over the others. A PSA unit is based on the fact that adsorbents can be easily regenerated by a pressure change. The more adsorbable components are removed from a gas mixture by adsorption at high pressures, PH. Subsequently, the adsorbed species are desorbed using low pressures, PL. The basic Skarstrom47 PSA cycle makes use of this concept. Two fixed-bed adsorbers are used, as shown schematically in Figure 1. Each bed undergoes a cyclic operation (see Figure 2; bed 1) involving four stages: (1) The first stage is pressurization (with the outlet sealed) to a pressure of PH, using stored gas (air in this study), fed in at a constant flow rate, Qpress, and for a time, tpress. (2) The second stage is adsorption of the more adsorbable components (N2 in this study), from a mixture (air) fed in at a constant pressure of PH, at a constant flow rate, Qfeed, and for a time, tads. A raffinate stream richer in the less adsorbable components (O2) is produced.

(3) The third stage is blowdown (reducing the pressure from PH to PL) with no inlet stream, but with a constant outlet flow rate, Qblow, of the extract for a time, tblow. (4) The fourth stage is purge with a part of the raffinate stream at a lower inlet pressure of about 1.1PL, flowing in at a constant rate, Qpurge, for a time, tpurge. The gas coming out of the purge stage goes as waste. Several researchers48,49 have studied the recycling of the waste gas into the feed stream. Such a recycling is not considered in this study to keep the analysis simple. Mathematical Model for a Two-Bed PSA Unit. A rigorous model3,27 has been developed3,50-52 by our group earlier, for a single fixed-bed adsorption column. Complete details of this model are available in ref 3. These comprise unsteady mass and energy balance equations for the macro-voids (gas “phase”), as well as mass balance equations for the micro-voids (inside the porous particle “phase”). It is assumed that no radial gradients are present in the bed and that the temperature and pressure inside a porous particle at the axial location, z, attains the values just outside of it (in the surrounding gas at z), instantaneously, at the same time, t. The gas mixture inside the micro-void in the particle at axial position, z, is assumed to be in local equilibrium with the carcass solid at the same radial position, r, and at time, t. The set of mass balance equations for the n components in the macro-void and the low Reynolds’ number Ergun equation55 relating the pressure gradient, ∂P/∂z, to the superficial velocity, Vs, are not all independent. They are related through the ideal gas law which gives the pressure, P, in terms of the total concentration, cbt (sum of the n concentrations), in the gas. The PDE for cbt is simplified using the ideal gas equation, and the Ergun equation is then substituted to get a second-order (in z) PDE for P. This is given in eq AI.2. The final equation for P, the equations for the n - 1 components in the macro-void, and the n equations for the micro-void are given in Appendix I. These comprise a complete set (with several correlations given in ref 3). This rigorous model3 has advantages over the more popular LDF model in that slightly better (nonoscillatory) results are obtained with fewer finite-difference grid points. These equations also apply to each of the four stages (for any one bed) of a PSA unit, provided different initial and boundary conditions (IC/BCs), as appropriate, are used to integrate them. The latter are given in Table 6. This procedure is similar to that followed by earlier workers,53 who used the LDF model instead of the more detailed model used here.3 The PDEs in Appendix I that characterize the gas phase (macro-void) are converted into a set of ordinary differential equations (ODEs) using the finite difference technique in the z direction. Fifty (ng) equally spaced finite-difference grid points are used. Central difference formulas,54 accurate to O[(∆z)2], are used for the internal grid points. Single-sided formulas are used for the two end points, z ) 0 and z ) L, where L is the length of each bed. The resulting set of ODEs (in t) for the macro-void is of the form:

dx ) f(x, Np*, qp, cp, vs) dt

(1)

The PDEs in Appendix I describing the particle (micro-void) are converted to ODEs using the method of orthogonal collocation54 (OC) in the radial direction, r, at any z and t. Three OC points are taken in the particles at r ) r1, r2, and r3. These lead to a set of ODEs of the form

dcp ) f(cp, qp) dt

(2)

Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3753

Figure 2. Operating principle of the basic Skarstrom47 PSA cycle. The operation at any stage of a cycle is shown, as are also the corresponding flow rates and time intervals for the four stages.

In eqs 1 and 2 b xT ≡ [cb1(1), cb1(2), ..., cb1(ng + 1), ..., cn-1 (ng + 1); P(1), P(2), ..., P(ng + 1); T(1), T(2), ..., T(ng + 1)] p/ p/ p/ (Np*)T ≡ [Np/ 1,r(1), N1,r(2), ..., N1,r(ng + 1), ..., Nn,r(ng + 1)]

(cp)T ≡ [cp1(r1, 1), cp1(r2, 1), cp1(r3, 1), cp1(r1, 2), ..., cpn(r3, ng + 1)] (qp)T ≡ [qp1(r1, 1), qp1(r2, 1), qp1(r3, 1), qp1(r1, 2), ..., qpn(r3, ng + 1)] (vs)T ≡ [Vs(1), Vs(2), ..., Vs(ng + 1)]

(3)

Clearly, the ODEs for the macro-void are coupled to the equations for the particle (micro-void) because of cp, qp, and Np*. The OC expressions for derivatives are used to obtain Np*, while that for integrals is used to obtain the radially averaged h ads (required in eq AI.5a). values, qp, cp, and R The complete set of coupled ODEs, obtained after using the finite difference technique and the OC technique, as described above, are integrated numerically using the Petzold-Gear54,56 differential-algebraic equation (DAE) solver. Code DASPG from the IMSL library is used. The use of a DAE solver, instead of a differential equation solver like Gear’s method,54 is necessary

for this system because of the boundary conditions in the adsorption and purging stages at z ) L (Table 6), which leads to algebraic equations at z ) L. The PSA is operated in the following order of stages: 1, 2, 3, and 4 (see Process Description). Because the optimization involves objective functions and constraints under the CSS, we need to obtain results under this condition. We do this by assuming some starting conditions and repeating the computations for a few cycles until the CSS is attained. We assume that, at the beginning (first cycle), bed 1 is saturated with the feed gas (air) at PL and 298 K. This means that the entire macrovoids and the micro-voids are filled with this air [but that the carcass solid inside the particles is at a concentration that is in equilibrium (at a different composition) with this air, with values given by the Freundlich-Langmuir isotherms]. Similarly, at the beginning of the operation, bed 2 is assumed to be saturated with the more adsorbable component, N2, at PH and 298 K. This means that the entire macro-voids and the micro-voids are filled with this N2 [but that the concentration of N2 in the carcass solid inside the particles is that given by the FreundlichLangmuir equation]. Thereafter, the initial conditions of the two beds for any stage are taken as the final conditions at the corresponding previous stages. The computations are carried out until the CSS is attained. It is also assumed that the initial concentration of the gaseous components inside the porous particle is equal to the concentration in the macro-void at the same r, z, and t. At t g 0, the desired gas (or mixture of gases) is fed in at z ) 0 or withdrawn at z ) L (z is assumed to be in the direction of flow and so changes direction periodically), at

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Table 1. Variables/Parameters Used for the PSA Unit (O2-N2-Zeolite 5A) variables/parameters

Table 3. Comparison of the Results Obtained from the Tuned Model and the Experimental Data Used for Tuning

value

(a) Details of the PSA length of the adsorption bed, L inside diameter of adsorption bed, D adsorbent shape pellet, Dp, jr bed porosity, b pellet porosity, p porous pellet density, Fp Qblow Qpress

Unit53,57 0.35 m 3.5 × 10-2 m sphere 0.0707 × 10-2 m, 40 Å 0.4 0.4 1160 kg/m3 115.395 × 10-6 m3/s 57.7 × 10-6 m3/s

(b) Parameters and Properties Compiled3 from the Literature specific heat of oxygen, Cp1 29.2 kJ/(kmol K) specific heat of nitrogen, Cp2 26.4 kJ/(kmol K) / 920.48 kJ/(kg K) specific heat of “solid”, CpS effective thermal conductivity of gas, ke 1.2254 kJ/(m h K) 2.930 kmol/m3 saturation constant for O2, qp10 4.120 kmol/m3 saturation constant for N2, qp20 pre-exponential coefficient for the 1.325 × 10-4 equilibrium constant of O2, k10 pre-exponential coefficient for the 4.859 × 10-6 equilibrium constant of N2, k20 heat of adsorption of O2, (∆Hads)1 -5.508 × 103 kJ/kmol heat of adsorption of N2, (∆Hads)2 -16.378 × 103 kJ/kmol Ueff 0.0 A1 A2 C1 C2 τp n1 n2

(c) Tuned (First-Level) Parameters3 0.500 1.500 1.622 2.000 4.298 1.022 1.018

B1 B2

(d) Tuned (Second-Level) Parameters (This Work) 0.577 1.300

Table 2. Bounds for the Decision Variables in Equations 5-7 decision variable (units)

lower bound

upper bound

Qfeed (m3/s) tads (s) Qpurge (m3/s) tpurge (s) PH (atm) PL (atm)

19.23 × 10-6 20 5.77 × 10-6 20 2 1

57.69 × 10-6 90 38.47 × 10-6 60 4 2

corresponding pressures and temperatures. Table 6 summarizes these conditions. Optimization Problems Studied. The most interesting and relevant problems in any multi-objective optimization study involve more than two objective functions. However, it is difficult to attempt and understand the results if we start with considering too many objective functions. Simple problems involving only two objectives need to be studied first so as to build insights and get a good “feel” of the terrain of the objective function space. A brief discussion of some simple as well as some more complex multi-objective optimization problems for PSAs is now described. These are mere samples of the immense number of possibilities that exist for optimization. In this paper, we consider the multi-objective optimization of the operation of an existing (operating) PSA unit, the same as described by Farooq et al.57 Details of this unit are given in Table 1. A possible set of decision variables for this unit is Qfeed, tads, Qpurge, tpurge, PH and PL. It is assumed that (reasonable) values of Qpress and Qblow are specified53 and that

simulation

experiment57

70

76

8

12

purity of O2 (%) in the accumulated raffinate recovery of O2 (%) in the accumulated raffinate

Table 4. SA Parameters Used in MOSA-aJG parameter Nseed Pjump S(0) T

value

parameter

value

0.888 76 0.5 0.5 50

NB,i ()NT,i)

100; i ) 1 10; i ) 2, 4-6 5; i ) 3 2

fb

these are not additional decision variables (at least in the optimization problems considered here). Using these values of Qpress and Qblow and the randomly generated values of PH and PL (in any solution), values of tpress and tblow (required for computing the objective functions described later) may easily be computed without any trial and error (the opposite calculation, e.g., calculating Qpress and Qblow from PH, PL, tpress and tblow, requires a cumbersome trial and error procedure). MOSAaJG is used to obtain the optimal solutions for the multiobjective problems described below using these six decision variables. The objective functions could be selected from among several possibilities: the purity of the products in the raffinate, Ra, accumulated over time, tads, and in the extract, Ex, accumulated over tblow and the recovery of the individual components in the (accumulated) raffinate and in the (accumulated) extract, all under CSS conditions. These are given, mathematically, by

f1 ≡ purity of O2 in accumulated Ra ≡

∫0t

∫0t

CORa2 dt

ads

CORa2 dt +

ads

∫0t

CNRa2 dt

ads

(4a)

f2 ≡ purity of N2 in the accumulated Ex ≡

∫0t

∫0t

CNEx2 dt

blow

CNEx2 dt +

blow

∫0t

COEx2 dt

blow

(4b)

f3 ≡ recovery of O2 in accumulated Ra ≡

∫0t

∫0t

QRaCORa2 dt

ads

QRaCORa2 dt +

ads

∫0t

QRaCNRa2 dt

ads

(4c)

f4 ≡ recovery of N2 in accumulated Ex ≡

∫0t

∫0t

QExCNEx2 dt

blow

QExCNEx2 dt +

blow

∫0t

QExCOEx2 dt

blow

(4d)

An important constraint that needs to be satisfied is that the total (accumulated in one stage) number of moles of gas, Nads,out (see nomenclature for definition), collected as raffinate in the adsorption stage should be larger than the total number of moles of gas, Npurge,in, used for purging. This will ensure that there is a net production of purified gas from the unit. This is required


Figure 3. Plots of the non-dominated solutions obtained for Problem 1 (eq 5) after different number of simulations.

since the decision variables, Qfeed and Qpurge, are generated randomly.

Three sample multi-objective optimization problems are now formulated:

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Figure 4. (a) Set of non-dominated solutions and (b-g) the decision variables for Problem 1 (eq 5), after 150 CSS simulations; 9, experimental point.


Problem 1

Table 5. Decision Variables and Objective Functions for Points 1-3 in Figure 4a

max f1(Qfeed, tads, Qpurge, tpurge, PH, PL) max f3(Qfeed, tads, Qpurge, tpurge, PH, PL)

(5a)

problem 1 (eq 5) non-dominated solution

(5b) decision variable

subject to (s. t.):

Constraints Nads,out > Npurge,in

(5c)

model equations (Appendix I)

(5d)

Bounds on the decision variables see Table 2

(5e)

Problem 2 max f2(Qfeed, tads, Qpurge, tpurge, PH, PL)

(6a)

max f4(Qfeed, tads, Qpurge, tpurge, PH, PL)

(6b)

subject to (s. t.):

Constraints and bounds eqs 5 c-e

(6c)


(7a)


(7b)


(7c)


(7d)

Problem 3

subject to (s. t.):

Constraints and bounds eqs 5 c-e

(7e)

Results and Discussion In the present study, MOSA-aJG,1,2 with some modifications and with periodic interaction, is used to solve the three multiobjective optimization problems described in eqs 5-7. A summary of the modified algorithm is provided in Appendix II. The modifications and the interactive steps used help obtain the Pareto optimal solutions more rapidly in this computationally intensive problem. It was observed from our earlier studies1,2 that the number of function evaluations required to obtain the Pareto set using SA-based algorithms is lower than that for GA for a few benchmark problems as well as for a computationally intensive real-life problem in Chemical Engineering. Hence, (a modified version of) MOSA-aJG is used in this study to obtain the solutions. A computer code is written for the simulation and multi-objective optimization of the two-bed four-stage PSA unit in MS Fortran Power Station. The CPU time required for simulation (to attain CSS, for any set of decision variables) on a Pentium 4 (2.99 GHz) is 24 h. The large amount of the CPU time is due to the large amounts of CPU time taken by the blowdown stage. The CPU time required for solving any of the two- and four-objective optimization problems (eqs 5-7) on a Pentium 4 (2.99 GHz) is 720 h with this algorithm. Several parameters (and properties) need to be specified for the PSA unit53,57 described at the beginning of Table 1. Many

purity (%) of O2 in Ra (f1) recovery (%) of O2 in Ra (f3) Qfeed (m3/s) tads (s) Qpurge (m3/s) tpurge (s) PH (atm) PL (atm)

point 1

point 2

point 3

34.1

54.1

75.8

88.8

63.5

38.3

36.73 × 10-6 62.14 33.2 × 10-6 25.2 3.81 1.13

46.16 × 10-6 25.0 31.43 × 10-6 22.6 3.58 1.065

42.7 × 10-6 27.5 19.85 × 10-6 46.9 3.13 1.018

of these are compiled from the literature (see subheading b in Table 1). Seven additional parameters have been tuned3 (referred to as first-level tuning) using one set of experimental data of Jee at al.58 on the same system on a single bed (single stage only). Two parameters, B1 and B2, have been tuned (secondlevel tuning) using one set of experimental data57 on a PSA unit. The experimental conditions for this set are as follows: feed flow rate (Qfeed) ) 25 × 10-6 m3/s; average flow rate of raffinate product, QRa ) 1.13 × 10-6 m3/s; tpress ) 60 s; tads ) 40 s; tblow ) 60 s; tpurge ) 40 s; PH ) 3.0 atm; and PL ) 1.0 atm. The values of all the other parameters (except the two being tuned) are taken to be the same as given in Table 1. A reasonably good agreement is observed between the predictions of the tuned (after the second-level) model and the data57 used for tuning. This is shown in Table 3. The tuned model is then used to optimize this PSA unit using the modified MOSA-aJG. The values of the upper and lower bounds used for the six decision variables are listed in Table 2. The computational parameters required for the adapted MOSA-aJG for solving Problems 1-3 are given in Table 4. These parameters are chosen on the basis of the experience of earlier studies1,2 on the optimization of different industrial chemical engineering systems. Problem 1 (Equation 5). Because these problems are computationally intensive, the standard algorithm for MOSAaJG1,2 takes extremely large CPU times to converge to a good set of non-dominated solutions. Hence, the multi-objective optimizations were carried out with a modified version of the original algorithm with periodic interactions, as described in Appendix II. The set of non-dominated solutions obtained after NT,1 ) 100 simulations for Problem 1 is shown in Figure 3a. Because a large gap is observed after point 1, this point is selected as a base point for further perturbation. Ten (NB,2) additional simulations are then carried out. The improved set of non-dominated solutions is shown in Figure 3b (after a total of 110 simulations). At this stage, it is observed that point 2 should be selected for a similar operation. Again, with this as a new base point, five (NB,3) further simulations are carried out. The non-dominated solutions at the end of each of the several such operations are shown in Figure 3, along with the total number of simulations and the new base point to be selected. It is found that the adapted MOSA-aJG leads to a reasonably good set of non-dominated solutions much more rapidly than does the original algorithm.1,2 The final set of non-dominated solutions (Figure 4a, the same as Figure 3g) and the associated decision variables for Problem 1 (eq 5) are shown in Figures 4a-g. The experimental values of the objective functions and the corresponding decision variables (in the absence of any optimization) are also shown in Figures 4a-g so as to indicate the improvement obtained by optimization. Good improvement in the recovery of O2 in the raffinate and a slight improvement

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Figure 5. (a) Set of non-dominated solutions, (b-g) the decision variables, and (h) the purity of O2 in raffinate for Problem 2 (eq 6).

in the purity of O2 in the raffinate from that of the experimental point are observed. If we go from any one point (say, 1) to

another (say, 2) in Figure 4a, we find that f1 (purity of O2 in Ra) improves (increases) whereas f3 (recovery of O2 in Ra)


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Figure 6. (a-d) Sets of non-dominated solutions and (e-j) the decision variables for Problem 3 (eq 7); 9, experimental point.

worsens (decreases). This plot, therefore, represents a set of nondominated solutions. A decision maker would have to decide on the “preferred” solution (operating point) from among these several solutions. Table 5 reports the values of the decision variables for three non-dominated solutions, points 1, 2, and 3, in Figure 4a. Figure 4b-g shows the decision variables corresponding to the set of non-dominated solutions in Figure 4a. It is observed from Figure 4a-c,f that tads decreases, while PH and Qfeed increase in the left-hand end of the non-dominated set. Because of PH and tads, more of the nitrogen gets adsorbed and the exiting gas is purer in oxygen. This region, however, is also associated with increasing values of Qfeed. The effect of the latter predominates in determining the recovery of oxygen, which is found to decrease. In the same region of the non-dominated set, it is observed that tpurge and PL decrease, while Qpurge increases. The effects of PL and Qpurge dominate, and more of the adsorbed nitrogen is flushed out, giving a bed that is leaner in nitrogen at the start of the adsorption stage. This also leads to more adsorption of nitrogen in the adsorption stage and purer oxygen in the exiting gas. A complex interplay of the several decision variables is, thus, observed. Three computational parameters, namely, NB,i, T, and NT,i, were varied, one by one, around their reference values used for generating the results shown in Figure 3g. The results, after 150 simulations, were observed to be the best for the reference values of the parameters. Results are not being shown here but can be made available on request. Problem 2 (Equation 6). The set of non-dominated solutions (Figure 5a) and the corresponding decision variables (Figure 5a-g) obtained for this problem are shown in Figure 5. It may be emphasized here that in the adapted algorithm, all the solutions are stored until NT,1 (taken as 100) simulations, irrespective of the choice of the objective functions. The solutions stored in File 1 from Problem 2 are, therefore, identical to those in Problem 1, saving considerable CPU time (the plot of the non-dominated solutions in File 2 at the end of NT,1 simulations will differ from that in Figure 3a, because different objective functions are selected for Problem 2 than for Problem 1). The periodic interaction procedure is followed only after this point, and two different objective functions are used hereafter. The interaction procedure used for this problem is the same as for Problem 1. The final set of non-dominating solutions obtained after 150 simulations is shown in Figure 5a. It is observed from Figure 5a that the ranges of f2 and f4 are much smaller than those (f1 and f3) obtained for O2 in Figure 4. Figure 5b-g shows the decision variables corresponding to the set of non-dominated solutions of Figure 5a. The effects of feed

(adsorption) flow rate (Qfeed) and the purge time (tpurge) are seen to be sensitive (Figure 5b,e) compared to those of the other variables. It is observed from Figure 5b,e that the purity of N2 in the accumulated extract increases while its recovery reduces as Qfeed and tpurge increase. The increase in Qfeed (for almost the same tads; see Figure 5c) increases the amount of N2 adsorbed and leads to an increase in the purity of O2 in the accumulated raffinate (see Figure 5h). This subsequently leads to a higher purity of N2 in the accumulated extract during the blowdown operation. The increase in tpurge (purging with high purity O2; Figure 5h) at almost constant Qpurge increases the removal of the highly adsorbed component, N2. This ensures higher adsorption of N2 in the following adsorption stage, which, in turn, leads to the production of higher purities of N2. Unfortunately, experimental values (without optimization) of the two objective functions are not available. Problem 3 (Equation 7). With the physical insights developed on Problems 1 and 2, we can proceed to solve the more difficult, four-objective problem (eq 7). The set of nondominated solutions is shown in Figure 6a-d. Because fourdimensional (for the four objective functions) plots cannot be made, f1 is selected as a “reference” objective function and the solutions are put in order of increasing values of f1. Figure 6a shows the values of f1 as a function of the re-classified solution numbers. The remaining three objective functions for these solution numbers are shown in Figure 6b-d. The corresponding decision variables are given in Figure 6e-j. Experimental values (without optimization) are also plotted (after the largest value of the solution number) where available. It is observed that while f1 increases (improves) continuously, f3 (worsens) decreases. This is sufficient to classify the solutions as a non-dominated set. The third objective function, f2, the purity of N2 in the accumulated extract, is observed (Figure 6c) to be almost constant. The objective function f4 increases at the beginning but decreases thereafter. So it is non-dominated with respect to f3 in a certain domain but is otherwise in the remaining range. This kind of behavior of solutions being non-dominated with respect to each other only in part of their ranges is quite interesting. The associated decision variables are plotted in Figure 6e-j. A much larger scatter is observed for this problem, though some general trends can be discerned. The large scatter in problems with three and four objective functions has been observed in our earlier studies too. The trends observed for the behavior of the decision variables in this problem (Figure 6e-j) are similar to those in Problem 1. So, similar physical explanations are applicable.


IC/BCs:

Conclusions

t ) 0: cpi (r, z, t ) 0) ) cbi (z, t ) 0)

(AI.3b)

r ) Rp: cpi (r ) Rp, z, t) ) cbi (z, t)

(AI.3c)

The rigorous model3,27 developed for a single fixed-bed adsorption column is modified for use in solving the different stages (pressurization, adsorption, blowdown, and purge) of a PSA unit, by changing the boundary and initial conditions. The model is tuned using one set of experimental data available in the literature.57 Three multi-objective (two two-objective and one four-objective) optimization problems are formulated on the operation of a PSA unit. An adaptation of MOSA-aJG (with periodic interactions) is used to obtain sets of non-dominated solutions for the PSA unit, using up to four objective functions with constraints. Detailed explanations on the effects of the decision variables on the different objectives are provided. The procedures developed (for simulation and optimization) are quite general and can be applied to other PSAs of industrial importance.

Dusty gas model:

Acknowledgment

e Effective Knudsen Diffusivity (Dk,i ):

Partial financial support from the Department of Science and Technology, Government of India, New Delhi [through Grant SR/S3/CE/46/2005-SERC-Engg, dated November 29, 2005], and from the Ministry of Human Resource Development [through Grant F.26-11/2004.TS.V, dated March 31, 2005], Government of India, New Delhi, are gratefully acknowledged.

-Deff i

r ) 0:

p/ Ni,r

)

[

∂cpi )0 ∂r

∂cpi ∂r

1

n

1

+

∑ j*i

e ) 97.0 × 0.36 × 10-6 jr Dk,i

e ) Di,j

(

Macro-void (gas phase):

∑ i)1

n

| [

]

b

r)Rp

∂2cbi bDL 2

) 0; i ) 1, 2, ..., n - 1 (AI.1)

∂z

Total pressure: ∂P

P ∂T

2

∂P

T ∂t

∂t

+ PK

( )( )

2

( ) ( )( ) ∂P

-

KP ∂P ∂T

∑

DL

[ ( )( ) 2 ∂P ∂T

+

T ∂z ∂z

|

P ∂2T T ∂z2

-

( )]

2P ∂T 2

[

[ ]

)(

∂t n

kj e(-∆H ∑ j)1 0

n

∑ j)1 j*i

0

ads)j/RT

(AI.4b)

T1.5

0.5

P ϑ 2Ω 98.665 i,j Di,j (AI.4c)

)

]

∑

2

(AI.2)

T ∂z

qpi (z,t) )

3 Rp3

R 2 p r qi (r, z, t) dr ∫r)0

(AI.5b)

cpi (z, t) )

3 Rp3

R 2 p r ci (r, z, t) dr ∫r)0

(AI.5c)

p

p

∂qpi 3 R h ads,i ) ) 3 ∂t R p

kj e(-∆H ∑ j)1

p τp

with

n

) (1 +

0.5

∂ ∂ bCpi (cbi T) + p(1 - b)Cpi (cpi T) + ∂t ∂t ∂ ∂T (qpi T) + Fs(1 - b)(1 - p)Cp/S ) (1 - b)(1 - p)Cpads i ∂t ∂t n ∂ ∂T ∂ ke - Cpi (Vscbi T) + (1 - b)(1 - p) × ∂z ∂z ∂z i)1 n 4 (-∆Hads)iR h ads,i + Ueff(TJ - T) (AI.5a) D i)1

Intraparticle diffusion: ∂cpi

T Mw,i

∑

+ ∂z T ∂z ∂z ∂z n 3(1 - b) ∂K ∂P ∂2P p/ + RT Ni,r + DL P ∂z ∂z r)Rp i)1 Rpb ∂z2

)

+K

2

( )

p 1 1 0.36 × 0.001 858 + p M M τ w,i w,j

Energy balance:

3 ∂2 P ∂P ∂ci - Kcbi 2 (1 - b) - K Rp ∂z ∂z ∂z

]

(AI.4a)

e Effective molecular diffusivity (Di,j ):

Model Equations3 of a PSA Unit. The following is a summary of the final equations.

∂cbi p/ - Ni,r ∂t

ypj

e 1 - ypi j)1 Di,j

e Dk,i

Appendix I

b

(AI.3d)

cpj )2[p(1 +

p Rp 2∂qi r r)0 ∂t

∫

dr

(AI.5d)

The IC/BCs for the differential equations for cbi , P, and T for the different stages of PSA are given in Table 6.

ads)j|/RT

cpj )2 + qip0(1 - p)(ni(cpi )(ni-1)ki0e(-∆Hads)i/RT +

nj(cpj )(2nj-1)(kj0e(-∆Hads)j/RT)2)]-1

[ ( )] 1 ∂

r2 ∂r

r2Deff i

∂cpi ∂r

(AI.3a)

Appendix II Modified Version of MOSA-JG/aJG. Step 1. Generation of the first NT,1 ()100) points Set counter: l ) 0; l is the total number of simulations (feasible as well as non-feasible)

3762


Table 6. IC/BCs for the Four Stages of PSAa t)0

stage no. (for bed 1) 1 (pressurization)

concentration

pressure

temperature

2 (adsorption)

concentration

pressure

temperature

3 (blowdown)

concentration

pressure

temperature

4 (purge)

concentration

pressure

temperature

az

t ) 0) ) specified (see text) P(z, t ) 0) ) specified (see text) T(z, t ) 0) ) specified (see text) cbi (z, t ) 0) ) final profile after pressurization P(z, t ) 0) ) final profile after pressurization T(z, t ) 0) ) final profile after pressurization cbi (z, t ) 0) ) final profile after adsorption P(z, t ) 0) ) final profile after adsorption T(z, t ) 0) ) final profile after adsorption cbi (z, t ) 0) ) final profile after blowdown P(z, t ) 0) ) final profile after blowdown T(z, t ) 0) ) final profile after blowdown

cbi (z,

z)0

( )| ( )| ( )| ( )| ( )| ( )|

-bDL

∂cbi ∂z

Vs ) -K -bke

∂T ∂z

-bDL

Vs ) -K -bke

∂T ∂z

) cbi CpiVs(T|z)0- - T|z)0+)

∂T(z ) L, t) )0 ∂z

) Vs(cbi |z)0- - cbi |z)0+)

∂cbi (z ) L, t) )0 ∂z

z)0

∂P ∂z

∂P(z ) L, t) )0 ∂t

z)0

z)0

∂cbi (z ) L, t) )0 ∂z ∂P(z ) L, t) )0 ∂z

z)0

z)0

∂cbi ∂z

) Vs(cbi |z)0- - cbi |z)0+)

z)0

∂P ∂z

z)L


∂T(z ) L, t) )0 ∂z

∂cbi (z ) 0, t) )0 ∂z

∂cbi (z ) L, t) )0 ∂z

∂P(z ) 0, t) )0 ∂z

Vs ) -K

∂T(z ) 0, t) )0 ∂z

∂T(z ) L, t) )0 ∂z

( )| ( )| ( )|

-bDL

∂cbi ∂z

z)0

∂P Vs ) -K ∂z -bke

∂T ∂z

z)0

) Vs(cbi |z)0- - cbi |z)0+)

(∂P∂z )|

z)L

∂cbi (z ) L, t) )0 ∂z ∂P(z ) L, t) )0 ∂t

z)0


∂T(z ) L, t) )0 ∂z

is taken in the direction of flow of the gas stream (and flips over for any bed after every two stages).

Step a: Generate a point within the bounds of X using several random numbers, (RN)i: lfl+1 U X(l) i ) Xi,L + (RN)i (Xi - Xi,L); 0 e (RN)i e 1; i ) 1, 2, ..., Nd Do jumping gene operation (see Step A below) on X(l) go to step a; continue until l ) NT,1 Step 2. Check for feasibility Set counter l ) 0 Set counter: kf ) 0; kf is the number of feasible points (satisfying all the constraints) Step b: l f l + 1 Take the lth point from the NT,1 points generated in Step 1 Check for the constraints; if not feasible, kf is not updated (kf) (kf) If feasible, kf f kf + 1; XFS ) X(l) (XFS is the kfth feasible point); calculate (kf) f) f (k i (XFS ); i ) 1, 2, ..., m; store in File 1 Go to step b; continue until l ) NT,1 Step 3. Check for non-dominance Set kf ) 1; set counter kd ) 1; kd is the number of nondominated points (1) (1) (1) Store X(1) ND ) XFS ; fND,i ) fi , i ) 1, 2, ..., m; in File 2 of non-dominated points Step c: kf ) kf + 1 (kf) Check for non-dominance of XFS (see Step B below) Go to step c; continue until all the feasible points in File 1 are checked. Print results and wait for external input of a selected base point, XB.

Step 4. Exploration about a selected base point Note: The first return to a base point is done after NB,1 [≡NT,1] total simulations. Thereafter, returns to base are after NB,i further simulations, i ) 2, 3, ... (NB,i ) 10; i ) 2, 4, 5, 6, ...; NB,3 ) 5) (acc) ) 0; (acc) is the number of accepted feasible points (that are worse than the previous point) (acc) If l ) ∑NB,i, X(l) ) XB; X(acc) AC ) XB; fAC,i (XB); i ) 1, 2, ..., m (acc) UAC,i )

[

(acc) XAC,i - Xi,L

(XUi - Xi,L)/2

]

- 1; i ) 1, 2, ..., Nd

(acc) this gives -1 e UAC,i e1 (l) (acc) Ui ) UAC,i Step d: l ) l + 1 Generate a new point, U(l), using (l-1) U(l) + (RN)iS(acc) ; -1 e (RN)i e 1; i ) 1, 2, ..., Nd; i ) Ui i (acc) initial Si ) 0.5 Convert U to X using (l) U X(l) i ) {[(Ui + 1)/2][Xi - Xi,L]} + Xi,L; i ) 1, 2, ..., Nd Do the jumping gene operation (See Step A) on X(l) Check feasibility (bounds; constraints): If not feasible, go to Step d (kf) (kf) f) If feasible, kf f kf + 1; XFS ) X(l); calculate f(k i (XFS ); i ) 1, 2, ..., m. Store in File 1 Check for acceptance (see Step C below) go to Step d; continue until the next printing: l ) ∑NB,i


Check for the non-dominance of each of these newly accepted points, X(i) AC; i ) 1, 2, ..., acc, one by one (see Step B) After this, set X(i) AC ) 0 for i ) 1, 2, ..., acc Step e: Annealing schedule is called after every ∑NB,i simulations after NT,1. Ti is updated using Ti ) 0.9Ti Print all solutions in File 2 and wait for the next input of a selected base point go to Step 4 Step A. Jumping gene operation Choose one of the two operations: either random JG adaptation (JG) or fixed-length JG adaptation (aJG) (a) Random-length JG adaptation Generate two random numbers Convert to integers, p1 and p2, between 0 and Nd Replace the decision variables so identified using U X(l) i ) Xi,L + (RN)i(Xi - Xi,L); i ) p1 to p2; 0 e (RN)i e 1 (b) Fixed-length JG adaptation (aJG) Generate one random number Convert to an integer, p, between 0 and Nd - fb Replace all the decision variables between p and p + fb using U X(l) i ) Xi,L + (RN)i(Xi - Xi,L); i ) p to p + fb; 0 e (RN)i e 1 Step B. Check for non-dominance based on selected fi’s (kf) (Assume XFS as X(i) AC in the following operation for l > NT,1) (kf) Compare XFS with eVery non-dominated member present, j ) 1, 2, ..., kd, in File 2, one by one (kf) If XFS is a non-dominated point, copy it in File 2: kd ) kd (kf) (kd) (kf) d) + 1; X(k ND ) XFS and fND,i ) fi ; i ) 1, 2, ..., m; return (kf) If XFS is dominated over by any member already present in File 2, do not include it in File 2; return If any of the earlier members, j (j ) 1, 2, ..., kd), in File 2 are (kf) dominated by XFS , delete all such X(j) ND from File 2 and re-number the solutions in this file using the following procedure: (j) Make X(j) ND ) 0; fND,i ) 0; i ) 1, 2, ..., m; for all j dominated (kf) over by XFS Set counter k ) 0 and j ) 0 Step f: j ) j + 1 (k) (j) (k) (j) if X(j) ND * 0; k ) k + 1; XND1 ) XND and f ND1,i ) f ND,i; i ) 1, 2, ..., m go to step f; continue until j ) kd Re-number solutions in File 2 (with zeros at the end, to give kd solutions) (k1) (k1) (k1) 1) X(k ND ) XND1 and f ND,i ) f ND1,i; i ) 1, 2, ..., m; for k1 ) 1, 2, ..., k (kf) (k+1) (kf) X(k+1) ND ) XFS ; f ND,i ) fi ; i ) 1, 2, ..., m (k1) (k1) XND ) 0 and f ND,i ) 0 for k1 ) k + 2 to kd kd ) k + 1; return Step C. Acceptance of a feasible point (l > NT,1) in File 1 (kf) Calculate the probability, P[XFS ], of acceptance of the new feasible point m

f) P[X(k FS ]

)

exp ∏ i)1

[

]

(acc) f) -(f (k i - f AC,i )

Ti

; (initial Ti ) 50)

(kf) (kf) (kf) ] > 1, XFS is superior to X(acc) If P[XFS AC , accept XFS (kf) (kf) If 0 < P[XFS ] < 1, XFS is worse than the previous point, (kf) (kf) X(acc) AC ; accept XFS with a probability of P[XFS ] (kf) (kf) If XFS is accepted; (acc) ) (acc) + 1; X(acc) AC ) XFS ; (acc) calculate fAC,i ; i ) 1, 2, ..., m;

Calculate (acc) UAC,i

)

[

(acc) XAC,i - Xi,L

(XUi - Xi,L)/2

]

- 1; i ) 1, 2, ..., Nd

U(l) ) U(acc) AC Update S(acc) using i (acc) (acc) (acc-1) (acc-1) Si ) 0.9Si + 0.21[UAC,i - UAC,i ]; s.t.: 1 × 10-4 (acc) e Si e 0.5 and U(l) Else, no update of S(acc) i Nomenclature Ai ) tuning parameters (≡ ki,used/ki,literature; i ) 1, 2, ..., n) Bi ) tuning parameters (≡ qip0,used/qip0,literature; i ) 1, 2, ..., n) cbi ) molar concentration of component i in the gas “phase” (macro-void) at any z, t, kmol (m3 of macro-void)-1 cbt ) total molar concentration in the gas “phase” (macro-void) at any z, t, kmol (m3 of macro-void)-1 cpi ) molar concentration of component i in the micro-void at any r, z, t, kmol (m3 of micro-void)-1 cpi ) spatially averaged molar concentration of component i in the micro-void at any z, t, kmol (m3 of micro-void)-1 eff eff Ci ) tuning parameters (≡ Di,used /Di,estimated ; i ) 1, 2, ..., n) Cpi ) specific heat of component i in the gas, kJ kmol-1 K-1 Cpads ) specific heat of component i in the adsorbed phase i (carcass solid), kJ kmol-1 K-1 Cp/S ) specific heat of the carcass solid, kJ kg-1 K-1 D ) internal diameter of the column, m Deff i ) effective diffusivity of component i in the micro-void gas, m2 h-1 e Di,j ) effective binary diffusivity of component i in component j, m2 h-1 Dek,i ) effective Knudsen diffusivity of component i in the micro-void gas, m2 h-1 DL ) axial diffusion coefficient of the gas mixture in the macrovoid, m2 h-1 fb ) fixed number of decision variables used in the adapted MOSA-aJG fi ) ith objective function ke ) effective thermal conductivity of the bed, kJ h-1 m-1 K-1 ki ) adsorption equilibrium constant of component i, m3 kmol-1 ki0 ) pre-exponential coefficient for ki, m3 kmol-1 K ) coefficient in the Blake-Kozeny equation in the gas “phase” (macro-void) at any z, t, m2 kPa-1 h-1 [≡ Vs/(-∂P/∂z)] L ) length of the fixed-bed adsorber, m m ) number of objective functions Mw,i ) molecular weight of component i, kg kmol-1 n ) total number of components ni ) Langmuir-Freundlich parameter (eq AI.3a) for component i ng ) number of equally spaced finite-difference grid points in the bed Nads,out ) number of moles of gas collected (accumulated over tads) at the exit of the bed in the adsorption stage, kmol NB,1 ) number of simulations to be performed for the first return to base operation () NT,1) NB,i ) number of additional simulations to be performed before return to base; i ) 2, 3, ...

3764


Nd ) number of decision variables p/ ) flux of component i based on the superficial (carcass Ni,r solid plus micro-void) area of the porous particle at any r, z, t, kmol i (m2 superficial area)-1 h-1 Npurge,in ) number of moles of gas going inside the bed (during tpurge) in the purging stage, kmol Nseed ) random seed used in random number generator NT,1 ) number of simulations to be performed for the first return to base operation NT,i ) the number of simulations to be performed before reducing the temperature; i ) 2, 3, ... P ) total pressure of the gas mixture at any z, t, kPa PH ) adsorption pressure in the PSA operation, kPa PL ) desorption pressure in the PSA operation, kPa Pjump ) jumping gene probability qpi ) concentration of component i on the carcass solid at any r, z, t, kmol (m3 of carcass solid)-1 qip0 ) saturation constant for the adsorbed component i on the carcass solid at any r, z, t, kmol m-3 qpi ) average concentration of the adsorbed component i on the carcass solid at any z, t, kmol (m3 of carcass solid)-1 Qblow ) flow rate of gas coming out from the bed undergoing blowdown, m3/s Qfeed ) flow rate of gas entering into the bed undergoing adsorption, m3/s Qout ) flow rate of the gas mixture exiting from the column at time t during the adsorption stage, m3/s Qpress ) flow rate of gas entering into the bed undergoing pressurization, m3/s Qpurge ) flow rate of gas entering into the bed undergoing purging, m3/s QRa ) average raffinate product flow rate [≡ 1/tads{∫t0adsQout dt - Qpurgetpurge}], m3/s r ) radial position in the particle, m jr ) average radius of pores in the micro-void, m R ) universal gas constant, kPa m3 kmol-1 K-1 R h ads,i ) net average rate of adsorption of component i on to the carcass solid at any z, t, kmol (m3 carcass solid)-1 h-1 Rp ) radius of the porous pellet, m S(k) ) Nd-dimensional vector of step sizes in the kth acceptance t ) time, h tads ) adsorption stage time, s tblow ) blowdown stage time, s tpress ) pressurization stage time, s tpurge ) purging stage time, s T ) temperature of the gas mixture in the macro- and microvoids at z, t, K T ) m-dimensional vector of computational temperatures in MOSA TJ ) jacket fluid temperature, K Ueff ) overall heat transfer coefficient from the jacket fluid to the inside gas, based on the inside area of the column, kJ m-2 K-1 h-1 Vs ) superficial velocity in the gas “phase” at any z, t, m h-1 X ) Nd-dimensional vector of decision variables, Xi ypi ) mole fraction of component i in the gas in the micro-void z ) axial position in the bed, m Greek Symbols b ) void fraction in the gas “phase” (macro-void) p ) void fraction in the adsorbent particle (micro-void)

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ReceiVed for reView November 28, 2006 ReVised manuscript receiVed February 19, 2007 Accepted March 8, 2007 IE0615180

Multi-Objective Optimization of Pressure Swing ... - ACS Publications

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