The Optimal Control of a Periodic Adsorber:

., Natural Convetion in nonnewtonian Fluids Heated from Below, paper presented at First Paifi Chem. Eng. Congr. (1972). Samuels, M. R., Stability of a Fluid in a Long Horizontal, Retangular Cylinder Heated from Below, Ph.D. thesis, Univ. Mihigan, Ann Arbor ( 1966).., and S. W. Churhill, Stability of a Fluid in a Retangular Region Heated from Below, AChE J., 13, 77 (1967). Shmidt, E., and P. L. Silveston, Natural Convetion in Horizontal Liquid Layers, Chem. Eng. Progr. Symp. Ser. No. 29,55,163 ( 1959). Thomas, R. W., Finite Differene Computation of Heat Trans fer by Natural Convetion, Ph.D. thesis, Univ. New South Wales, Australia (1970). Tien, C., H. S. Tsuei, and Z. S. Sun, Thermal nstability of a Horizontal Layer of nonnewtonian Fluid Heated from Below, ntern. J. Heat Mass Transfer, 12, 1173 ( 1969). Tsuei, H. S., Thermal nstability and Heat Transport of a Layer of nonnewtonian Fluid, Ph.D. thesis, Syrause Univ. New York (1970). Wikes, J. O., and S. W. Churhill, The FiniteDifferene Computation of Natural Convetion in a Retangular Enlosure, AChE J., 12,161 (1966). Manusript reeived November 11, 1971; revision reeived June 21, 1972; paper aepted July 10, 1972. The Optimal Control of a Periodi Adsorber: DANEL E. KOWLER and ROBERT H. KADLEC Deportment of Chemial Engineering The University of Mihigon, Ann Arbor, Mihigon 481 04 Part 1. Experiment Cyli pressure variations in a fixed bed adsorber an ause signifiant separation of gaseous mixtures. Feed pressure hanges are the driving fore in this parametri pumping proess. The optimal feed sequene is maximum (maximum flow in) pressure, zero flow (variable pressure), maximum flow out (minimum pressure). For the nitrogenmethane feed gas at 168 k N/m2 and a 1.22 m bed of adsorbent, the optimal yle time is 3 seonds, and feed is sustained 5v0 of the time. The zero flow mode is unneessary if produt purity is the sole objetive, The omponents of a binary gas mixture exhibit different adsorption harateristis on a solid adsorbent suh as a moleular sieve, the differene usually being expressed as a relative volatility. f suh a mixture is steadily passed through a fixed bed of adsorbent partiles, the adsorbed phase will adjust to a omposition whih is onsistent with the equilibrium adsorption isotherms for the system for the given feed omposition and the bed temperature. Under suh a steady flow ondition, no separation of the mixture ours after the adsorbent is loaded. f the feed pressure is alternated between values higher and lower than the bed entrane pressure, a feed flow will alternate with feedend exhausting of the bed. This an be done while ontinuously withdrawing gas at the produt end of the bed. f suh yling is ontinued until eah yle is like the previous, the produt gas is enrihed in the least strongly adsorbed omponent. The system will not separate if run exlusively at the maximum feed rate or at the minimum (negative) feed rate, thus an optimum feedexhaust poliy must exist. Correspondene onerning this paper should be addressed to R. H. Kadle. SCOPE Furthermore, if the yle time is too short, the bed sees a mean inlet feed pressure, and no separation ours. Likewise, an infinitely slow yle time means steady operation, and again, no separation. Thus an optimum yle time exist;. The objetives of the optimization an be either produt purity or apaity, or a ombination of both. The sope of Part of this paper is the determination of the experimental optimum pressure wave form and yle time for the methanenitrogenmoleular sieve system. The theory of Part 1 indiates that the family of square waves enompasses the optima for a system in dynami equilibrium. Therefore, the variables of yle time, yo of time at maximum feed pressure, no feed flow, and minimum (exhaust) feed pressure were experimentally explored. Other quantities, suh as bed length and diameter, feed omposition, temperature and moleular sieve parameters, were not varied. The model of this system was previously developed for the in:tantaneous equilibrium assumption and is employed here. The theoretial optimization proedures were both an analysis of the model partial differential equations and an extension of the proedure of Horn and Lin, based on the Pontryagin Maximum Priniple. Hene, the sope of this work inludes the experimental AlChE Journal (Vol. 18, No. 6) November, 1972 Page 1207

and theoretial optimization of the yli adsorber, via the use of a simplified model. The results are ompared with experiment, and the potential effets of unvaried parameters are predited using dimensional analysis. The potential appliations of the methods inlude the optimization of any distributed parameter yli proess. The optimum feed pressure wave form was found to be a square wave, for the objetive of produt purity alone. f apaity is inluded in the objetive, a period of no feed and no exhaust must be inluded. The yle time for maximum separation is approximately 3 seonds, at approximately 507' ontime, for the nitrogenmethane system in a laboratorysale devie. Thus, the optimal operating sequene is maximum feed, no feed or exhaust, maximum exhaust. The theoretial work provided an exellent guide to the experimental by narrowing the sope of the experimental work. Computations whih were very inaurate omposition preditors, due to large finite differene step size, were very aurate optimization routines. We onlude that the odtima are insensitive to the Droess model. These alulatioas are exeedingly omple; and should be at CONCLUSONS AND SGNFCANCE tempted only if there are large inentives for the optimization. The operability of this unsteady proess was easily implemented with automati onoff valves; no diffiulties in ontrol of the proess were experiened. A steady flow of produt was obtained. No moving parts were replaed in over a 100 hours of operation, but valve wear would be expeted in longer periods of operation. t is signifiant that this proess has no steady state analog and therefore annot be desribed against the bakground of a steady state theory. Joined with other parametri pumping proesses, a whole new spetrum of proesses appear for potential use by industry. This separation proess has now been quantified and optimized for one partiular equipment. For most ontinuous proesses within the hemial industry, it is assumed that using onstant operating onditions is the best mode of operation. The reent studies of Wilhelm ( 1966, 1968), Douglas ( 1967), and Shrodt (1967) have shown, however, that yli or periodi operation in some proesses an lead to proess improvements. Sine the magnitude and diretion of the hange in proess outputs hanges with yle type, it is important to loate the best possible operating onditions. This requires both experimentation and the appliation of optimal ontrol theory. An inherently periodi proess and the subjet of this researh is a ylially operated moleular sieve adsorber. An outgrowth of Skarstrom's (1959) heatless adsorber, this gas separation olumn has pratial signifiane due to the following advantages over some present adsorption methods: 1. No separate adsorbent regeneration proess neessary 2. Continuous operation 3. No solids handling needed during operation. n addition, its fast startup time may make it pratial where the startup delays of onventional ontinuous mode plants are not aeptable. This yli adsorption proess, as illustrated in Figure 1, is operated at ambient temperature by alternating between a flow of feed gas mixture and a flow of exhaust gas at one end of the olumn, while regulating the produt end of the olumn for onstant flow. The manner in whih the olumn is pressurized and depressurized at the feed end (feed boundary pressure ontrol) greatly affets the omposition of the produt stream and the amount of gas Feed Gas ~ Moleular Sieve Bed Pressure n W Fig. 1. Periodi adsorption proess. Produt exhausted. Thus, it is desirable to establish the optimal feed boundary pressure ontrol whih maximizes ertain performane riteria of the adsorber. One performane index whih provides for minimizing exhaust rate as well as maximizing produt omposition is given by 1 + 1 = 4 (Composition C [Exhaust rate]) dt (1) 7 where 7 is the period of a yle and where the value of the onstant C determines the importane of the minimization of exhaust rate relative to the importane of the maximization of produt omposition. The modeling of this adsorption proess, the appliation of optimal ontrol theory, and the aompanying numerial analysis will be presented in Part 1 of this paper. The adsorber an be desribed by two partial differential equations for pressure and omposition, and an ordinary differential equation at the produt boundary. The neessary onditions for the optimal ontrol problem have been derived both for the distributedparameter system and for a ell model approximation of it. Computational work using the ell model and experimental studies were used to maximize the performane index for a fixed produt flow rate and a limited available pressure for the feed gas. This artile will deal with the experimental optimization of separation of the binary nitrogenmethane mixture. With unlimited possible ontrol poliies, an exhaustive experimental exploration of all proess variables would be unrealistially omplex. Therefore, the atual adsorber performane was investigated for variation in a general optimal pressure ontrol form that may be established from the appliation of optimal ontrol theory. These theoretial studies have shown that the general optimal pressure ontrol form, whih an also be simply desribed as the optima1 feed boundary flow ontrol, should be of the form [maximum flow in, zero flow, minimum flow in (whih is maximum flow out) 1. The experimental investigation was further simplified by onsidering a performane index whih onsiders only the maximization of produt omposition. Page 1208 November, 1972 AlChE Journal (Vol. 18, No. 6)

THE EXPERMENTAL ADSORPTON SYSTEM The basi experimental adsorption system used in this researh was based upon the work of Turnok and Kadle (1971) and is shown in Figure 2. The desired yli pressure ontrol forms an be physially onstruted with two 2way solenoid valves using the following yling sequene: 1. Feed valve open, exhaust valve losed 2. Both valves losed 3. Feed valve losed, exhaust valve open These valves are eletrially ontrolled by relays whih are ativated by the square wave outputs of an analog omputer. Surge vessels are used for both the inlet and outlet flow. The nitrogenmethane mixture, pressure regulated at the feed gas ylinder, passes through a 8.2 dm3 surge vessel, and the exhaust flows through a 36.0 dm3 surge vessel before flow measurement, All other flow resistanes are kept to a minimum. The solenoid valves were hosen with a 6.35 mm internal orifie, whih presents very little resistane to flow. The only other obstale to flow between the surge vessel and the adsorption olumn is one thikness of 80 mesh sreen used to support the adsorbent. n his researh, Turnok found that the use of rushed moleular sieve pellets led to variable flow resistane aused by attrition of the adsorbent partiles. Use of 2050 mesh round partiles of moleular sieve as the adsorbent eliminated that problem. Adsorbent is paked into a 1.9m diameter, 1.52m long, steam jaketed, shedule 40 pipe. This paked bed is mounted on a 35" angle to eliminate the possibility of forming a void hannel along the bed. At the produt end of the olumn, the exit stream passes through a pressure regulator whih regulates the pressure of the downstream flow. The flow to the produt flowmeters is nearly onstant whereas the flow at the end of the olumn varies signifiantly. This produt stream is then metered by two rotameters: one metering and ontrolling the flow to a thermal ondutivity ell; the other measuring and ontrolling the remainder of the produt flow. The omposition of this produt stream is measured by the alibrated thermal ondutivity ell. A 100% N2 stream, also serving as jhe referene flow, a 50% N250% CH, mixture, and the feea gas mixture were used as the alibration gases. The output signal from this thermal ondutivity ell is displayed on a digital voltmeter. A total material balane on the operating system an be ompleted when the exhaust flow is measured. The flow pulses from the olumn are damped by the large surge volume so that the flow is easily measured by two wet test meters. To ahieve the desired yli pressure ontrol, the signals ativating the relays whih ontrol the feed and exhaust valves must be properly set. Care must be taken to make ertain that the losing of the exhaust valve exatly oinides with the opening of the feed valve. f there is any time that both valves are simultaneously open, gas will bypass the olumn and flow diretly from the feed to the exhaust. The feed pressure is limited with the pressure regulator on the feed gas ylinder and the produt flow is adjusted with the produt rotameters. The feed pressure was 168 kn/mz for this study. During the startup period, the omposition is ontinuously metered with the output of the thermal ondutivity ell. By reheking the output of this thermal ondutivity ell for the alibration gases before and after eah run, the relative unertainty of a omposition is within 0.2%. queny, 0.35 yles/seond, at whih the maximum in produt omposition ours. Thus the variation in the timing of the valve swithes was arried out at only one frequeny for all produt flow rate studied. Next, the exhaust flow rate is examined as a funtion of the frequeny of the applied ontrol. Figure 4 shows that the faster the ontrol is yled, the more gas is exhausted. This behavior led to a omputational study that inluded exhaust rate minimization in the performane index. The exhaust rates are only slightly sensitive to the hanges in produt flow rate; the exhaust rate dereases slightly as the produt flow rate is inreased. At the onstant frequeny of 0.35 yles/seond, the effet of the length of time that the feed valve is open on the produt omposition is shown in Figure 5. As antiipated, a lear maximum in this omposition is exhibited. t is lear that if the valve were either not open at all or Vent Referene Gas Thermal Condutivity Cell 2Way Solenoid Valves n Referene Gas Rolameters 00Calibralion Gas Rotameter Feed Gas Pressure Water Saturator 0: Fig. 2. Shemati diogrom of experimental adsorption system. Vent 6 RESULTS The initial experimental studies were run with the yli sequene (maximum flow, minimum flow), that is, the feed and exhaust valves alternately opening. Figure 3 presents the effet of frequeny on produt omposition. t is noted that neither a different produt flow rate, a variation in the fration of the period that the feed valve remains open (FFVO), nor a small hange in feed gas omposition from 28.6% N2 to 32.2% Nz hange the fre AlChE Journal (Vol. 18, No. 6) 0 2 4 6 8 10 PEROD, Seonds Fig. 3. Effet of frequeny on produt omposition. November, 1972 Page 1209

if it were open 100% of the time, no steady state separation would be possible. For simple onoff operation of the feed and exhaust valves, the ontrols to maximize produt omposition for different produt flow rates have now been experimentally speified. The effet of delaying the opening of the exhaust valve after the feed valve has been losed is shown in Figures 6 and 7. DSCUSSON OF RESULTS The experimental results show that the ontrol omponent of zero flow is not needed in the ontrol sequene to maximize produt omposition. The optimal frequeny is 0.35 yles/seond and there is no notieable effet of hanging produt flow rate on this frequeny. n addition, it an be seen from Figure 5 that the optimal fration of the period spent applying maximum pressure inreases slightly with inreasing produt flow rate. By inluding a separate term for exhaust minimization in the performane index, as well as produt omposition maximization, the ontrol omponent of zero flow beomes important. This omponent is applied after the maximum pressure omponent is applied. t appears that applying the zero flow ontrol for a short interval (< 6% of the period) does not notieably derease the produt omposition although it does signifiantly redue the exhaust rate and thus improves the performane of the adsorption system. To onstrut this ontrol omponent, two valves were used at the feed boundary of the adsorption olumn. The timing of the optimal sequene [feed valve open (exhaust valve losed), both valves losed, exhaust valve open (feed valve losed)] will depend upon the relative importane of the exhaust minimization term ompared to the term for maximization of produt omposition. n the thermal parametri pumping separation proess Fig. 5. Effet of fration of period feed valve open on produt omposition for 0.35 yles per seond and 32.2%N2 feed. 1 "E 200 V 42 W a CZ 5 250," 150 Lo 3 Q L! 100 50 0 EXPERMENTAL RESULTS % of Period Feed Valve 0 35 2 4 6 8 10 PEROD, Seonds Fig. 4 Effet of frequeny on exhaust flow rate for a produt flow rate of 9.14 st.m3/s. Page 1210 November, 1972 0 E V : W a 5 s LL n = a X W st.mj / s 110 0 9.14 A 14.16 0 18.90 100 0 4 8 10 PER CENT OF PEROD BOTH VALVES CLOSED Fig. 6. Effet of losing both valves on exhaust flow rote (0.35 ps, feed 32.2% Nz, and feed valve open 47% of period). AlChE Journal (VOl. 18, No* 6,

65 L2 z" 611 e EXPERMENTAL RESULTS Produt %of Period flowrate Frequeny feed Valve ~ (st.rn3ls) (ps) _ Open 0 9.14 0.35 Varied 9.14 Varied 50 6 14.16 0.35 Varied A 14.16 Varied 55 0 18.90 Varied 49r 45i 4 a 12 PER CENT OF PEROD BOTH VALVES CLOSED Fig. 7. Effet of losing both valves on produt omposition. (0.35 ps, feed 32.2% N2, and feed valve open 47% of period.) developed by Wilhelm et al. (1966, 1968), the perfonnane depends upon alternating the diretion of the flow of the solvent. t is this kind of behavior in the yli adsorber that gives rise to the possibility of the optimal ontrol omponent of zero flow. Thus, it is antiipated that this ontrol omponent may also play a role in the optimal ontrol of the thermal parametri pumping proess. A full theoretial and numerial investigation is neessary to define the atual optimal sequene. Figure 3 shows that the optimal frequeny does not notieably vary with produt rate. At a onstant frequeny of 0.35 yles/seond, whih orresponds to the maxima in produt ompositions in Figure 5, FFVO was varied for the flow rates of 9.14, 14.16, and 18.9 st.m3/s. Another fator of interest, for whih no experimental exploration was made, is the level of the available feed pressure. Although the separation for inreased pressure markedly inreases, the amount of gas exhausted inreases as well. n onsidering the pressure level for olumn operations, the inreased separations for higher pressure levels must be balaned by the inrease in the energy required for the pressurizing of the feed gas and the dereased fration of feed gas reovered as produt. A study, either numerial or experimental, would be required to find the optimal feed pressure. n the preeding experimental work a system of fixed dimensions was operated at different levels of produt throughput for a fixed available feed pressure. The problem that should now be onsidered is how these operating variables and system dimensions affet the operation of the yli adsorption proess. From the experimental results in Figures 3 and 5 it is apparent that dereased produt flow rate inreases the separation aomplished. However, aompanying this inrease is a derease in the fration of feed gas that is reovered as produt. llustrated in Figure 8 is the hange of the proess outputs of separation aomplished and fration of feed gas reovered, as the frequeny of operation is varied or as the fration of the period that the feed valve is open, is varied. These hanes are shown for the three different produt flow rates of 9.14, 14.16, and 18.9 st. m3/s. From this figure it is seen that dereased frequeny results in an inrease of the fration of feed gas reovered while, as in Figure 3, the aomplished separa AlChE Journal (Vol. 18, No. 6) \\ '\ \ \ '\ \ h '\ '\ \\ ' b ' 0.06 0'08 0.10 0.12 0.14 0.16 FRACTON OF FEED GAS RATE RECOVERED AS PROGUCT Fig. 8. Comparison of adsorption system outputs with individual variations in frequeny of ontrol or in the fration of period that the feed valve is open, for a feed gas omposition of 32.2% N267.8% CH4. tion goes through a maximum. t is lear that variations of the frequeny from the optimal value do not derease the separation aomplished with respet to the fration of feed gas reovered, nearly as muh as variations of FFVO from its optimal value. Thus, if the system is to be run at different flow rates, the frequeny of operation need not be set as arefully as the fration of the period that the feed valve is to be open. f the separation aomplished and the fration of the feed gas reovered as produt were the only important fators in the operation of the system, it would appear from Figure 8 that operation at lower produt flow rates would result in the best performane. However, the apaity of the system must also be onsidered. For example, if the reovered fration is 0.09, a 56.6% N 2 produt would be ahieved for a produt flow rate of 9.14 st.m3/s whereas a produt omposition of only 55.8% Nz would be ahieved for a produt flowrate of 14.16 st.m3/s. Although a smaller separation would be obtained for the latter operating ondition, a 55% inrease in apaity would result. Thus, in order to find the best overall operating onditions for the adsorption system, the apaity, as well as the fration of feed gas reovered as produt, needs to be onsidered. The system dimensions were onstant during the ourse of the experimental work. n addition, the bed permeability was a fixed quantity. n order to design a yli adsorption system, the effet on operations of these equipment speifiations must be understood. The higher the permeability, the greater the apaity of the system provided that the appropriate higher frequenies are used. Consideration of this behavior for design purposes requires two notes of aution. First, if larger adsorbent partiles are used to ahieve higher permeability, the optimal operation will be at higher frequenies November, 1972 Page 1211

and higher flow rates, for whih adsorption rate limitations may beome signifiant. Seond, unless the permeability is onstant, the optimal timing of the ontrol sequene will vary and unless orretions for this are made in the ontrol, suboptimal operation will result. To avoid this problem, adsorbent partiles that resist abrasion and maintain a onstant flow resistane should be used. The round partiles used in this researh were found to be satisfatory. Another fator that affets the operation of the adsorber is the length of the olumn. Unlike most hemial proess equipment, dereased length inreases the apaity of this system (within ertain limits). To ahieve the same produt omposition for shorter lengths, higher frequenies are required. Sine the optimal frequeny inreases as the inverse of the square of the length (Kowler, 1969), shorter lengths would require faster operation. This result is valid for flow rates for whih adsorption rate limitations an be negleted. n fat, sine the optimal frequeny inreases so quikly as length dereases, the performane of the ontrolling solenoid valves may limit the ahievement of the optimal frequeny for shorter lengths. Despite these limitations, it is lear that attempts should be made to use shorter lengths of olumn to inrease apaity and derease equipment osts at the same time. Having found (theoretially) the optimal feed bound ary yli ontrol of (maximum pressure, zero flow, minimum pressure) and having gained a better understanding of the design parameters, a yli adsorption system an now be more properly designed. For the separation of gas mixtures for whih there exists an adsorbent with a high relative volatility, the yli adsorption proess may well be of ommerial value. LTERATURE CTED Douglas, J. M., Periodi Reator Operation, Znd. Eng. Chem. Proess Design Develop., 6, 43 (1967). Kowler, D. E., The Optimization of the Cyli Operation of a Moleular Sieve Adsorber, Ph.D. thesis, Univ. Mihigan (1969). Shrodt, V. N., et al. Plant Sale Study of Controlled Cyli Distillation, Chem. Eng. Si., 22, 759 ( 1967). Skarstrom, C. W. Use of Adsorption Phenomena in Automati PlantType Gas Analyzers, Ann. N. Y. Aademy Si., 72, 751 ( 1959). Turnok, P. H., and R. H. Kadle, Separation of Nitrogen and Methane via Periodi Adsorption, AChE J., 17, 335 (1971). Wilhelm, R. H., etal., Parametri Pumping: Dynami Priniple for Se arating Fluid Mixtures, nd. Eng. Chem. Fundumntu~, 5, 141 (1966). Wilhelm, R. H., et al., Parametri Pumping: Dynami Priniple for Separating Fluid Mixtures, ibid., 7,337 ( 1968). Part. Theory The fixed bed binary gas adsorber, when alternately fed and exhausted at one end, produes a purified produt from the other end. Coupled partial differential equations in pressure and omposition, representing total mass and omponent balanes with loal equilibrium, desribe the operation. The Maximum Priniple is applied to determine the optimal yli unsteady feed poliy for the balaned objetives of produt purity and quantity. The sequene (maximum feed, no flow, maximum exhaust) is optimal. The experimental optimum is lose to the alulated optimum. Dimensional analysis is used to determine parametri effets. This yli adsorption devie represents an example of a distributed parameter feeddriven unsteady proess. Experimental work has shown that there exist both an optimal frequeny and an optimal feed pressure program for ahieving the goal of separation of a binary gas mixture. This paper will onern itself with the development of the model for the system, appliation of optimal ontrol theory, and the numerial solution for the optimal feed boundary pressure yle. The details and results of the experimental study of this system have been presented in Part. MATHEMATCAL MODEL We onsider first the development of a mathematial model of the moleular sieve adsorber and the formulation of neessary onditions for optimal ontrol. The state variables of the system and the adjoint variables of the ontrol problem are governed by partial differential equations. Beause of the omplexity of these equations and of the omputational proedures involved, the omputer ost for a finely spaed finite differene solution of these equations is exessive. Therefore a lumpedparameter model (ell model) is also developed. A pitorial model is shown in Page 1212 November, 1972 Figure 1. Feed and exhaust an either alternate or they an alternate with intermediate shutoff periods. n establishing the bases for a model for the moleular sieve bed, Turnok and Kadle (1971) made the following assumptions and approximations: 1. deal gas behavior 2. Dary s Law representation of the gas flow 3. Visosity of the gas phase is omposition invariant 4. Plug flow onditions 5. At any instant, equilibrium exists between the gas phase and the adsorbed phase. 6. The effet of the heat of adsorption on the temperature profile will be negleted; isothermal operation is assumed. i1 i =2 Fig.. Model for the ell system. i =n AlChE Journal (Vol. 18, No. 6)