ABSTRACT SIMULATION OF DYNAMIC PRESSURE- Professor Timothy A. Barbari Department of Chemical Engineering

Transcription

1 ABSTRACT Title: SIMULATION OF DYNAMIC PRESSURE- SWING GAS SORPTION IN POLYMERS Heathe Jane St. Piee, Maste of Science, 2005 Diected By: Pofesso Timothy A. Babai Depatment of Chemical Engineeing A tanspot model was developed to simulate a dynamic pessue-swing soption pocess that sepaates binay gas mixtues using a packed bed of non-poous spheical polyme paticles. The model was solved numeically using eigenfunction expansion, and its accuacy veified by the analytical solution fo mass uptake fom a finite volume. Results show the pocess has a stong dependence on gas solubility. The magnitudes and diffeences in gas diffusivities have the geatest effect on detemining an optimal paticle adius, time to attain steady-state opeation, and oveall cycle time. Soption and tanspot paametes fo thee diffeent polyimides and one copolyimide wee used to detemine the degee of sepaation fo CO 2 /CH 4 and O 2 /N 2 binay gas mixtues. The sepaation esults fo this pocess compae favoably to those fo membane sepaation using the same polyme, and significantly impoved pefomance when a second stage is added to the pessue-swing pocess.

2 SIMULATION OF DYNAMIC PRESSURE-SWING GAS SORPTION IN POLYMERS by Heathe Jane St. Piee Thesis submitted to the Faculty of the Gaduate School of the Univesity of Mayland, College Pak, in patial fulfillment of the equiements fo the degee of Maste of Science 2005 Advisoy Committee: Pofesso Timothy A. Babai, Chai Associate Pofesso Raymond A. Adomaitis Associate Pofesso Pete Kofinas

3 Copyight by Heathe Jane St. Piee 2005

4 Acknowledgements I would like to thank my adviso, Pofesso Babai, fo all of his help and guidance duing the couse of this eseach. In the shot time I have woked fo him, I have leaned much fom the shaing of his knowledge and expeience. I would also like to thank my co-adviso, Pofesso Adomaitis fo the use of his Matlab code fo the simulations pesented in this wok. Even though I was not officially a student of his, he always had time to help, and I have leaned much fom woking with him as well. ii

5 Table of Contents Acknowledgements...ii List of Tables...iv List of Figues...v Chapte 1: Backgound...1 Chapte 2: Model Fomulation Pocess Desciption Continuity Equation and Bounday Conditions Eigenfunction Expansion Solution Pocedue...8 Chapte 3: Model Veification Compaison to Analytical Solution Compaison to Method of Finite Diffeences Compaison to Equilibium Calculations...15 Chapte 4: Model Results and Discussion Detemination of Optimal Sepaation Paametes Polyme Selection Specifications fo Cabon Dioxide/Methane Sepaation Compaison to Polyme Chaacteistics fo CO 2 /CH 4 Sepaation Compaison to Membane Technology fo CO 2 /CH 4 Sepaation Specifications fo Oxygen/Nitogen Sepaation Compaison to Polyme Chaacteistics fo O 2 /N 2 Sepaation Compaison to Membane Technology fo O 2 /N 2 Sepaation...46 Chapte 5: Conclusions...48 Appendix: Matlab Code fo Simulation...50 Bibliogaphy...55 iii

6 List of Tables Table 2.1: Pessue-swing soption pocess paametes... 6 Table 4.1: Diffusivities and solubilities of selected polyimides and copolyimides Table 4.2: Polyme sepaation compaison fo CO 2 /CH 4 gas mixtue.. 35 Table 4.3: Sepaation compaison fo CO 2 /CH 4 gas mixtue 38 Table 4.4: Sepaation compaison fo O 2 /N 2 gas mixtue. 46 iv

7 List of Figues Figue 2.1: Schematic of gas sepaation appaatus. 5 Figue 3.1: Plot of mass uptake vs. dimensionless time. 13 Figue 3.2: Plot of selectivity vs. dimensionless time 15 Figue 4.1: Stuctues of selected 6FDA polyimides and copolyimide. 23 Figue 4.2: Gas phase concentation atio fo CO 2 /CH 4 sepaation. 27 Figue 4.3: Mole faction of mixtue extacted afte soption and desoption 29 Figue 4.4: CO 2 /CH 4 initial soption pofiles fo 6FDA-ODA paticles. 30 Figue 4.5: CO 2 /CH 4 initial desoption pofiles fo 6FDA-ODA paticles. 31 Figue 4.6: CO 2 /CH 4 steady-state soption pofiles fo 6FDA-ODA paticles Figue 4.7: CO 2 /CH 4 steady-state desoption pofiles fo 6FDA-ODA. 33 Figue 4.8: CO 2 /CH 4 mass uptake in polyme paticles afte successive cycles.. 34 Figue 4.9: Schematic of simple membane sepaation pocess.. 36 Figue 4.10: Schematic of pessue-swing soption pocess with 2 nd stage.. 37 Figue 4.11: O 2 /N 2 initial soption pofiles fo copolyimide paticles Figue 4.12: O 2 /N 2 initial desoption pofiles fo copolyimide paticles.. 42 Figue 4.13: O 2 /N 2 steady-state soption pofiles fo copolyimide paticles.. 43 Figue 4.14: O 2 /N 2 steady state desoption pofiles fo copolyimide paticles. 44 Figue 4.15: O 2 /N 2 mass uptake in polyme paticles afte successive cycles.. 45 v

8 Chapte 1: Backgound Gas sepaation pocesses have evolved ove the yeas to continually impove component sepaation, as well as incease cost effectiveness and efficiency. Many of these techniques have included vaiations of membane sepaation, pessue-swing adsoption (PSA), liquid absoption, and cyogenic sepaation. These pocesses ae often economically compaed based on the ate of poduction of sepaated components and quality of sepaation. 1 Two sepaations of paticula commecial inteest include cabon dioxide/methane sepaation and ai (oxygen/nitogen) sepaation. Cuent CO 2 /CH 4 sepaation applications include biogas sepaation fom landfills o fams fo enegy use, natual gas sweetening whee CO 2 is emoved fom natual gas wells, o enhanced oil ecovey whee supecitical o nea citical CO 2 is pumped into oil wells to educe oil viscosity fo easie extaction. 2,3 Ai sepaation applications cuently in use include oxygen enichment fo combustion o medical needs, inet gas blanketing on some oil tankes, o nitogen blanketing fo shipping o stoing food. 2 These sepaations have been achieved by using polymeic membane mateials in addition to the moe taditional methods. 2 The economic choice of sepaation pocess fo both CO 2 /CH 4 and O 2 /N 2 is highly dependent on the scale of that pocess. In geneal, membanes have been favoed at smalle scales. 1 Sepaation using polyme membanes has become an effective means of achieving gas sepaation in ecent yeas. Typical polymes used in commecial 1

9 membane sepaation have been polysulfones, polycabonates and cellulose acetates. 3,4 Recently, focus has tuned to polyimides due to thei excellent popeties fo sepaation. Some of these polyimides have been fomed into hollow-fibe membanes, 5,6 but few ae commecially available because of thei cost and difficulty to manufactue. 2,4 The fomation of integally-asymmetic hollow fibes in itself is not a simple task. Sepaation is not detemined by the polyme popeties alone, but on how the fibes themselves ae pepaed to povide a defect-fee thin film with the pope oientation to maximize sepaation. 3,6,7 Ceating a polyme solution with the desied themodynamic and heological popeties to fom hollow fibes is anothe challenge associated with this pocess. 3 Once the fibes themselves ae constucted, thee is also the issue of bundling them togethe to be used as a sepaation device fo eithe boe o shell feed, depending on the application. 2 Lastly, an appopiate fibe length must be chosen to pohibit significant pessue dop in the boe of the hollow fibes. 2 Given this difficulty in poduction, an altenative method fo sepaation is poposed hee that utilizes the sepaation chaacteistics of highly selective polymes by foming them into dense paticles and using them in a packed bed. These paticles could potentially be fomed by spay dying a polyme solution, o in situations whee a solution cannot be fomed, by simply ginding the polyme into smalle paticles. The use of sobent polyme paticles was developed and modeled by Babai et al. 8 fo liquidliquid extaction in a well-mixed batch pocess. In this wok, the focus instead will be on gas sepaation. In this thesis, the sepaation of binay gas pais, CO 2 /CH 4 and O 2 /N 2, is modeled using a dynamic pessue-swing soption pocess with dense polyme paticles. Two 2

10 diffeent polyimides ae used fo each gas pai, and thei sepaation pefomance compaed to a simple membane pocess which utilizes the sepaation popeties of the same polyimide. Although this pocess does not utilize a steady-state flux, sepaation esults fom this model will be shown to be compaable to a membane sepaation pocess. 3

11 Chapte 2: Model Fomulation 2.1 Pocess Desciption As an altenative to membane sepaation, the gas sepaation pocess descibed hee uses dynamic pessue-swing soption. This single-stage pocess utilizes a sepaation bed consisting of a vessel packed with nonpoous polyme sphees of unifom adius; the schematic can be seen in Figue 2.1. The concept fo this pocess is based on the liquid batch extaction pocess developed by Babai et al. 8 Fo the development of this specific model, two gases will be sepaated based on thei diffeing solubility and diffusivity in the polyme phase. To take advantage of the diffeent diffusivities, the pocess is stopped pio to equilibium, given that at equilibium, only solubility diffeences dominate sepaation. The poposed sepaation pocess cycle is composed of thee basic steps: 1) The packed bed is chaged with a compessed gas mixtue to a set pessue with valves 2 and 3 closed. When the desied pessue is eached, valve 1 is closed. 2) The gas in the extenal phase is sobed by the polyme paticles. At the end of the soption step, the emaining extenal phase is puged by opening valve 2 and loweing the pessue to P 2 (taken to be 0 atm fo the simulations of this wok). Afte the emaining gas in the extenal phase has been expelled, valve 2 is closed. 4

12 3) Valve 3 is opened, and vacuum is applied to allow the mass in the paticles to desob. Afte a given time, valve 3 is closed, and the sepaation bed is echaged with the compessed gas mixtue (step 1). 2 P 2 = 0 atm x A2 x B2 P polyme paticle feed x A1 x B1 1 extenal phase P 3 = 0 atm 3 x A3 x B3 Figue 2.1. Schematic of the gas sepaation appaatus using polyme paticles of a unifom adius. A manifold is used on the feed to ensue a apid fill time. A few assumptions and pocess simplifications wee made to ensue an accuate estimation of the physical esults. Fist, a manifold is used to ensue a shot fill time elative to the soption time scale, so an initial extenal phase concentation could be detemined. Fo the puposes of the model, the soption time scale was between 22 and 130 seconds fo the specified polymes, making this a valid assumption. In addition, the pocess is assumed to be isothemal. Ideal gas behavio was also assumed, allowing no change in gas compessibility with pessue. The specific paametes used fo the simulations in this thesis ae listed in Table

13 Sepaation bed volume 1L Polyme phase volume faction (constant) 0.5 Paticle adius (unifom, constant) µm Tempeatue (isothemal) 35 C Feed pessue Desoption pessue 20 atm 0 atm Table 2.1. List of paametes used in this wok to model the dynamic pessue-swing soption sepaation pocess. 2.2 Continuity Equation and Bounday Conditions The continuity equation that accounts fo the mass tanspot of each species into and out of the polyme phase is as follows: C t 1 = Di 2 i 2 C i (2.1) whee is the adial position in the spheical polyme paticle, C i is the concentation of component i and D i is the diffusion coefficient of species i in the polyme. These equations assume that the diffusion coefficient fo each species is independent of concentation. It is also assumed that the diffusion coefficient fo mass tanspot into the paticles is the same fo mass tanspot out. At the stat of the initial cycle, the polyme paticles ae assumed to be fee of any gas, theefoe: at t = 0, fo all : C i = 0 6

14 In addition, thee is no flux at the plane of symmety, o cente of the spheical paticles: C i at = 0, fo all t: = 0 The final bounday condition allows fo a mass balance between the extenal and polyme phases, as this specific pocess involves mass uptake fom a finite extenal volume. This bounday condition is at the inteface between the polyme and extenal phase, whee the ate the gas leaves the extenal phase is equal to the ate the gas entes the polyme phase at the bounday: at = R, fo all t: Ve Ci 3 Ci DiV p S t = R i = R = R (2.2) whee S i is the patition (o solubility) coefficient of component i in the polyme phase, and V e and V p ae the extenal and polyme phase volumes, espectively. The patition coefficients used hee ae defined as the atio of the concentation in the paticle elative to the concentation in the extenal phase, and is assumed to be independent of concentation. The extenal (o gas) phase is assumed to be well-mixed at all times; theefoe, convective mass tansfe esistance at the polyme suface is negligible. This assumption is easonable because the diffusion coefficient in many polymes (and those used in this study) is on the ode of 10-7 to 10-8 cm 2 /s, while that in the gas phase is 10-1 cm 2 /s. In addition, the gas sobed into the polyme is assumed to have no swelling effects on the polyme, allowing the extenal and polyme phase volumes to be assumed constant. Given the opeating time scales of the expeiment, the densities of the polymes chosen, a polyme volume faction of 0.5, and an initial pessue of 20 atm, the maximum possible 7

15 incease in polyme mass if all of the gas was sobed by the polyme paticles is only 1.6%. These specific paametes will be discussed late in Chapte Eigenfunction Expansion Solution Pocedue The pocess developed hee is dynamic, and theefoe cannot be epesented o estimated by an equilibium analytical solution. The methods of finite diffeences and finite element can be used to solve the patial diffeential equation listed in the pevious section, but these methods do not account fo the entie volume of the paticle, no can they captue concentation pofile behavio at the paticle inteface. In addition, the stability of the finite diffeences and finite element solution is based on the size of the time step elative to the spatial step size. Eigenfunction expansion models the concentation pofiles ove the entie paticle adius, and its solution accuacy is only contolled by the tuncation numbe, o numbe of basis functions, theefoe it is the method of choice fo this simulation. The concentation pofiles in time and space can be appoximated as the summation of an infinite numbe of functions: i= 1 C(, t) = a ( t) ψ ( ) + f ( t) (2.3) i i whee a i (t) ae coefficients detemined fom the given initial condition and f(t) is the finite-volume, concentation bounday condition that vaies with time. The ψ () in Equation 2.3 ae the othogonal basis functions geneated by the non-tivial solutions to the Stum-Liouville equation ove 0 < < 1: i 8

16 1 2 d d 2 dψ = λψ d o 2 ψ = λ ψ (2.4) i i i subject to: dψ (0) a + bψ (0) = 0 d dψ (1) c + dψ (1) = 0 d whee λ i ae the eigenvalues of the basis functions. These basis functions fo the Stum- Liouville equation wee detemined using Matlab code witten by Adomaitis 9 and solved using Matlab SV Release13. The time deivative of this estimation fo the concentation pofiles can then be substituted into the oiginal consevation equation (Equation 2.1) which can be witten as: C& = D 2 C 2 a& ψ + f& = D a ψ (2.5) i i i= 1 i= 1 Substituting the Stum-Liouville equation (Equation 2.4) into Equation 2.5 and multiplying both sides of the esultant equation by ψ yields: j i i i= 1 a& ψ ψ i i j + f& ψ j = D i= 1 a λψ ψ i i i j Integating both sides of this equation ove the volume of the spheical paticles, and taking into account the othogonality of the basis functions, a elationship fo the coefficients can be detemined: a & j = Da λ f& I (2.6) j j j 9

17 10 whee j = 1,2,3 and = R j j d I 0 2 4π ψ Since f(t) is the bounday condition at the paticle suface, f & can be detemined using the flux bounday condition at the suface, solving Equation 2.2 fo the time deivative of the concentation at the bounday: R e p C D R V V S t R C f = = = 3 ), & ( & (2.7) Substituting the appoximation fo the concentation pofile solution (Equation 2.1) into Equation 2.7 gives: R i i i a K f = = = ψ 1 & (2.8) whee D R V V S K e p 3 = Now that the equations fo f & and a elationship between a& and a have been established, Equation 2.8 can be substituted into Equation 2.6 and placed into matixvecto fom to be solved by a linea odinay diffeential equations solve: = = = = = = = = = = = = = f a a a K K K K I D K I K I K I K I D K I K I K I K I D f a a a j R j R R R j j j R j R j R j R R R j R R j M L L M M M M M L L & & M & & ψ ψ ψ ψ λ ψ ψ ψ ψ λ ψ ψ ψ ψ λ

18 This system of odinay diffeential equations (ODE) was solved with Matlab using a linea ODE solve witten by Adomaitis. 9 Once the coefficients ( ai ) and concentation at the bounday (f) ae detemined fo each time step, this infomation is then substituted back into the oiginal equation fo the estimation of the concentation pofiles (Equation 2.3) used to model the pocess discussed in this thesis. Once the concentation pofiles ae detemined, they can be used to find the mass uptake in the polyme paticles. The numbe of moles of i in the polyme phase at a given time can be found by integating the coesponding concentation pofile ove the volume of a paticle, multiplied by the numbe of paticles, N p : R 2 n ( t) = 4π N C (, t) d (2.9) i p 0 This calculation is used to detemine the majoity of the esults in this wok, including the mass balance of each gas in the polyme and gas phases and the sepaation pefomance of the dynamic pessue-swing pocess descibed in this chapte. i 11

19 Chapte 3: Model Veification Due to the dynamic natue of the soption/desoption pocess modeled hee, it was impotant to veify the behavio of the mass tanspot model pesented in the Chapte 2. To affim the accuacy of the model, the numeical solution was compaed to two diffeent known solutions: an analytical solution and the solution to these patial diffeential equations using the method of finite diffeences. 3.1 Compaison to Analytical Solution The numeical solution was compaed to the esults of the analytical solution fo the time-dependent mass uptake of a spheical paticle in a well-mixed solution of limited volume found by Cank 10 and given as: n( t) n 2 6β ( β + 1) exp( q τ ) = 1 n= β + q 2 n β n 2 (3.1) whee the q n s ae the nontivial solutions to tan q n 3q n = (3.2) 3 + βq 2 n τ is a dimensionless time vaiable defined as Dt τ = (3.3) 2 R and β is the atio of the extenal mass of penetant to the polyme phase penetant mass, applying the patition coefficient S 12

20 β = SN p Ve 4 π R 3 3 whee N p is the numbe of polyme paticles in the system. Fo the puposes of the solution compaison, the extenal and polyme phase volumes wee taken to be equal; theefoe, β simplified to 1/S. The solutions to Equations 3.1 and 3.2 wee computed using Matlab and compaed to the numeical solution, as shown in Figue 3.1. Fo this compaison, values of S = 5, D = cm 2 /s, and R = cm wee used fo both n(t)/n τ 1/2 Figue 3.1. Plot of polyme mass uptake against the dimensionless time vaiable, τ 1/2, compaing the numeical solution to the analytical solution developed by Cank 10 shown in Equation

21 the analytical and numeical solutions. To obtain the best estimate fo the exact solution, the fist 5000 tems of the summation wee used. 250 basis functions wee used to compute the eigenfunction expansion solution. Figue 3.1 veifies the accuacy of the numeical solution when compaed to the exact solution, whee the lagest oot mean squaed eo obseved between the two solutions using 250 basis functions was (0.5% eo) at τ 1/2 = Compaison to Method of Finite Diffeences To futhe veify the accuacy of the numeical solution, the eigenfunction expansion solution was also compaed to the solution obtained by Babai et al. 8 who used the method of finite diffeences. To detemine an optimal time and polymeic paticle adius fo thei liquid-liquid batch sepaation pocess, Babai et al. 8 plotted component selectivity against the squae oot of the dimensionless time vaiable, τ, whee selectivity was defined as: ( n A / nb ) α A / B = (3.4) ( C / C ) Ae whee n i is the numbe of moles of component i in the polyme phase and C ie is the concentation of component i in the extenal phase. The esults ae compaed gaphically, in Figue 3.2. Values of S A = 5 and S B = 1 wee used fo both calculation methods. 250 basis functions wee used to find the eigenfunction expansion solution. The close compaison of these plots (Figue 3.2), in combination with the excellent compaison to the exact solution (Figue 3.1), veifies the accuacy of the eigenfunction expansion solution. Be 14

22 D A /D B = 10 αa/b D A /D B = 2 D A /D B = 1 τ 1/2 Figue 3.2. Plot of selectivity, α A/B, (Equation 3.4) vs. τ 1/2 (Equation 3.3) using eigenfunction expansion and finite diffeences. The solid lines ae the esults fom the eigenfunction expansion solution, and the points ae the esults found by Babai et al. 8 using the method of finite diffeences. 3.3 Compaison to Equilibium Calculations The final test fo accuacy compaed the calculations fom the model at equilibium conditions to known values of concentation o mass uptake at equilibium. To best undestand these compaisons, it is impotant to intoduce the ideal selectivity between components A and B (α * A/B), which is defined as: S D A A α A / B = (3.4) S BDB 15

23 whee S i is the solubility (o patition) coefficient and D i is the diffusivity of component i. Fo a membane, α * A/B is equal to the membane pemeability atio, P A /P B, whee the pemeability fo component i is: P = S D (3.5) i i i Fo this dynamic pocess, α * A/B physically epesents the atio of initial fluxes at the inteface = R. When looking at selectivity, fo shot time scales both the diffeence in diffusion coefficients (D A /D B ) and diffeence in solubilities (S A /S B ) play a significant ole. Fo long time scales, the diffusion coefficients no longe dominate, and selectivity appoaches an equilibium value of S A /S B. This behavio was obseved by Babai et al., 8 and is also seen in these model esults, shown in Figue 3.2. In Figue 3.2, the selectivity appoaches the solubility atio S A /S B = 5 fo all diffusivity atios at long time scales. To calculate the exact mass uptake in the polyme paticles at infinite time, a basic algebaic elation was developed fo a finite volume, whee the initial mass of component i is equal to the sum of the masses in the polyme and extenal phases at equilibium: C V = C V + C o ie e ie e ip V p (3.6) whee o C ie epesents the initial concentation in the extenal phase, and C ip epesents the concentation in the polyme phase. The solubility coefficient is defined as the atio of polyme phase to extenal phase concentation: Cip Si (3.7) C ie 16

24 Substituting this elation into equation (3.6) gives the concentation in the polyme phase at equilibium, and can be defined in tems of C ie º: C ip o SiCieVe = V + S V e i p (3.8) Since the equilibium mass uptake of component i can be detemined by: n 4 R 3 C N 3 i = ip p π (3.9) whee N p is the numbe of spheical polyme paticles, the mass uptake of a component at equilibium can then be found by: n i o SiCieVe = V + S V e i p N p 4 π R 3 3 (3.10) Equation 3.10 was used to detemine the exact value of the equilibium mass uptake fo each component fo the soption step of the pocess. To simulate equilibium conditions, the model time scale was chosen to be 10 4 seconds. Fo uns using two sets of paametes fo each of the two gases, the infinite mass uptake values matched those of the exact solution fo the soption step to fou decimal places. The desoption step solution was veified by assuming an infinite esevoi (open 30 3 volume, simulated asve / V p = 10 cm ) with an extenal pessue of 0 atm. The time scale used to simulate equilibium conditions was 10 4 seconds. The expected behavio at infinite time fo an infinite extenal phase would esult in the evacuation of all mass in the polyme paticles. Fo two diffeent sets of paametes fo each of the two gases, the final mass in the paticles was on the ode of mol, veifying the accuacy of the behavio of the desoption step. 17

25 Chapte 4: Model Results and Discussion 4.1 Detemination of Optimal Sepaation Paametes Once the accuacy of the model was veified, the next step was to detemine which paametes will povide the best indication fo gas sepaation potential, specifically fo cabon dioxide/methane sepaation and oxygen/nitogen sepaation. Since this pessue-swing sepaation pocess deals with many of the same concepts as membane sepaation, simila paametes wee taken into consideation. When selecting a polyme fo a membane, the main citeia ae membane stability, mechanical popeties, cost, and most impotantly, component selectivity (Equation 3.4). Fo a membane in steady-state opeation with a constant flux, the best inheent sepaation is obtained by maximizing α * A/B. Like membanes, component solubility and diffusivity ae also key factos fo the pessue-swing soption pocess discussed in this wok, but the optimal elationship is not as staightfowad. Since the goal is to maximize the sepaation of the two gases, it is impotant to find a polymeic mateial that has diffeent intemolecula and physical inteactions (such as size selectivity) with each of the gases. A high solubility and a high diffusion coefficient in a given polyme esult in highe mass uptake of a component at a given peiod of time compaed to one with low solubility and low diffusivity. The ideal polyme fo sepaation would sob only one gas and leave the othe in the extenal phase, but this is not always possible fo gases with small molecula volumes such as CO 2, CH 4, N 2 and O 2. 18

26 Since this pessue-swing pocess does not utilize a steady-state flux as does a membane pocess, ideal selectivity is not the sole facto to conside when maximizing sepaation. In Equation 3.4, selectivity can also be epesented as a atio of mole factions fo each phase (o fo a membane, pemeate and etentate), which does not take into account the elative amounts of mass in those phases. Fo example, in the model pesented hee, a vey small amount of total mass in the polyme could have a high mass atio (n A /n B ) due to component A having a vey high diffusivity and solubility, but a lage amount of mass could still emain in the extenal phase with a mass atio nea unity. This combination would still esult in a high selectivity based on the mass factions, but would not esult in good sepaation fo this pocess because the amount of mass in the polyme phase elative to the mass in the extenal phase is ignoed. Theefoe, the bette indicato to detemine the optimal length of the soption cycle is the extenal phase concentation atio (C B /C A ). Maximizing this atio povides the highest extent of sepaation in the fist outlet steam, whee, duing the soption step, the extenal phase is puged at the time of maximum sepaation. The optimal length of the desoption step is much moe difficult to detemine due to the fact that the sepaation bed is open to vacuum and effectively, an infinite extenal phase volume. The main eason fo applying vacuum is to minimize mass build-up in the polyme paticles pio to subsequent soption steps. In detemining a desoption time, it is impotant to conside both mass build-up as well as total cycle time. A long desoption step will allow moe mass to leave the paticle, but could esult in an inefficient sepaation pocess oveall due to an excessively long cycle time, esulting in low poduct output. Theefoe, desoption step times wee chosen based on the attainment of a 19

27 constant mass etention afte successive cycles and the minimization of the numbe of cycles befoe this steady-state was eached. While it is desiable to have one component with high solubility and diffusion coefficients as well as a high solubility atio, this sepaation pocess is not optimized when the atio of diffusion coefficients is lage. This behavio is moe ponounced when the soption and desoption times ae equal. The eason fo this is that a lage diffeence in diffusion coefficients lends itself to excessive mass buildup of the moe slowly diffusing species. Afte successive soption and desoption cycles, this accumulation is amplified. Simulation uns with a modeate diffeence in the diffusivities of the two species will still esult in sepaation, but will not cause mass buildup of the slowe species. The magnitudes of the ideal selectivity and diffusion coefficients also play a ole in polyme selection. Solubility and diffusivity atios wee held constant fo compaative uns, but thei magnitudes wee alteed to detemine thei individual contibutions to sepaation by this pocess. Fo all paamete combinations, paticle adius was held constant and the desoption time was twice that of the soption time to inhibit mass build up in the polyme phase. The soption time scale was detemined by maximizing the extenal phase concentation atio fo each set of polyme paametes. The fist compaison consideed the magnitude of the diffusivities while the diffusivity atio emained the same (D A /D B = 2). To isolate diffusion effects, the solubilities of both components wee set equal to 1. The esults showed nealy equivalent sepaation fo both the soption and desoption steps of the sepaation pocess, even afte the magnitudes of the diffusion coefficients wee inceased by a facto of 5. 20

28 Based on these calculations, it appeas that the magnitudes of the diffusion coefficients themselves do not play a majo ole in sepaation fo this pocess. Howeve, the absolute value of the diffusivity does detemine the optimal soption/desoption time scale and the polyme paticle adius, both of which ae impotant pactical factos to be consideed fo the application of this pocess. The second compaison was to detemine the effect of the magnitudes of the solubility coefficients, while keeping the solubility atio constant (S A /S B = 5). Fo these tests, the diffusion coefficients wee set equal to one anothe, and the soption time scale was detemined by maximizing the extenal concentation atio. The esults of this compaison showed the pocess has a stong dependence on the magnitude of the solubility coefficients. Fo a 50/50 mixtue, inceasing the magnitude of the solubilities by a facto of 5 inceased sepaation in the soption step by 12%, but educed the sepaation in the desoption step by 12%. Lowe solubilities lead to geate sepaation in the polyme paticle, and theefoe geate sepaation in the desoption step when most of the gas is emoved fom the polyme phase. The low solubilities also esulted in a 468s (7.8 min) incease in the oveall cycle time when the paticle adius was held constant. Lastly, a compaison of the magnitudes of the poduct of S i and D i wee made to detemine its effect on the sepaation ability of the pocess. Fo the puposes of this test, (S A D A )/(S B D B ) = 10. When the poduct, S A D A, was inceased by a facto of 25 (incease in diffusivity by a facto of 5 and solubility by a facto of 5), the sepaation at the end of the soption step was inceased by 14% and the sepaation at the end of the desoption step was educed by 10%. Although an incease in the diffusion coefficient alone shows little elative change in sepaation, the magnitude of the diffusion coefficient appeas to 21

29 play a mino ole in sepaation fo this pocess when coupled with solubility. The 25-fold incease in magnitude of the S A D A poduct also shotened the soption time scale by a facto of Based on these esults, the magnitudes of S A and S A D A have the geatest effect on achieving the best sepaation fo this pessue-swing soption pocess. High S A D A and S A esult in geate sepaation in the soption step, while low S A D A and S A esult in geate sepaation in the desoption step. Depending on the specific application, good sepaation can be achieved with low o high values of solubility and the poduct of solubility and diffusivity. The elative sizes of these values will affect whethe o not bette sepaation is attained in the steam ich in the moe soluble and faste-diffusing component (desoption step) o in the steam ich in the slowe-diffusing, less soluble species (soption step). The magnitudes of these values will also have a lage effect on detemining polyme paticle adius and soption/desoption time scale length. 4.2 Polyme Selection As peviously mentioned, impotant factos to conside when choosing a polyme fo a sepaation pocess ae the solubility, solubility atio, diffusivity, and diffusivity atio. Thee is a consideable amount of liteatue on the sepaation of CO 2 /CH 4 and O 2 /N 2 using many diffeent polymes as membane mateials, with polyimides poviding excellent sepaation popeties fo both sets of gases discussed hee. Polyimides show excellent mechanical stength as well tempeatue and chemical esistance. 3 Aomatic polyimides, such as 6FDA-ODA, 6FDA-DAF, 6FDA-p-DDS, and copolyimide 6FDA-duene/mPDA (80/20) have igid backbones, and thei oveall 22

30 6FDA ODA (a) [ ] n DAF (b) [ ] n p-dds (c) [ ] n (d) duene m-pda [ ] n [ ] m Figue 4.1. Chemical stuctues of the epeat units of the polyimides (a) 6FDA-ODA, (b) 6FDA-DAF, (c) 6FDA-p-DDS, and (d) 6FDA-duene/mPDA copolyme. stuctues can be seen in Figue 4.1. As a esult of this igidity, a fozen fee volume is fomed due to egula voids. 3 In addition, thee is a stuctual hindance to packing in the 6FDA polyimides 3,4,11 due to the bulky CF 3 goups, adding to the fee volume in the polyme. Fee volume is impotant in inceasing diffusivity, but an impotant balance must be eached between diffusivity and diffusivity selectivity (D A /D B ). Packing density also plays a ole in size selectivity, so ideally a polyme would have enough fee volume to allow high diffusivities, but a low enough packing density that the mateial can still be selective. Polyimides have geate diffusivity selectivity compaed to taditional 23

31 Polyme D CO2 D CO2 /D CH4 S CO2 S CO2 /S CH4 6FDA-ODA FDA-p-DDS Polyme D O2 D O2 /D N2 S O2 S O2 /S N2 6FDA-DAF FDA-duene/ mpda (80/20) Table 4.1. Diffusion coefficients and solubilities of gases in selected polymes at 35ºC and 10 atm (O 2 data is at 2 atm). Diffusion coefficients ae in units of 10-8 cm 2 /s. Solubility coefficients ae in dimensionless fom, calculated with an assumed compessibility of 1. Supescipts efe to the efeence numbes. membane mateials, such as polysulfone and polycabonate, because they possess a highe packing density. 4 These data can be seen in Table 4.1. Ideal membane pemeability fo each of these polymes can be found fo a given gas by Equation 3.5. These values will be used fo the pessue-swing soption simulation and membane calculations to be discussed late in the chapte. As stated ealie, component solubility is anothe impotant sepaation paamete fo this pocess. Since solubility is mainly influenced by intemolecula inteactions, aomatic polyimides ae of inteest because they have many functional goups available fo these inteactions. The polyimides shown in Figue 4.1 have electon-ich aeas, available to donate electons, and aeas of electon acceptos; these chaacteistics allow fo geate sepaation 4 between CO 2 /CH 4 and O 2 /N 2. Absent any intemolecula inteactions, the solubility coefficient was shown to incease with inceasing fee volume. 3 24

32 Due to the fact that CO 2 and CH 4 have vey small diffeences in kinetic diamete 14 (3.3Å and 3.8Å, espectively) the diffeence in solubility between the two components is mainly due to the inteactions with the polyme. Since CO 2 is a quadupole, it has stong quadupole-dipole inteactions with the pola cabonyl and CF 3 goups in the 6FDA polyimides. In compaison, CH 4 is a non-pola molecule, and engages in weake van de Waals inteactions with the non-pola goups in the polyme, esulting in a lowe elative solubility. Although helpful in enhancing diffeences in solubility, these molecula inteactions between polyimides and CO 2 can potentially have negative effects on the pocess, such as polyme swelling. This swelling has been shown by Wind et al. 15 to affect the long-tem diffusivity and selectivity of simila polyimides. The plasticize effect of CO 2 has been shown to cause an incease in CO 2 diffusivity ove time. 15 It is not the pupose of this wok to model this time-dependent effect, but it is an impotant facto to conside in futue applications fo CO 2 sepaation. Due to thei simila espective kinetic diametes 14 of 3.46 and 3.64, O 2 and N 2 have typically been vey difficult to sepaate using polymes. A small diffeence in citical tempeatue and condensability between the two molecules esults in diffeing solubilities in polyimides; 3, 13 this diffeence shows the potential fo the use of polyimides as ai sepaation membanes. Based on the sepaation paametes discussed ealie, it is impotant to weigh optimal sepaation with pactical factos such as paticle adius, pocess time scale, and time to each steady-state. High diffusivities will esult in bette sepaation in the soption step and will lead to a shote time scale length; theefoe, the magnitudes of 25

33 these coefficients ae impotant. Also as impotant is the atio of diffusivities. A high diffusion coefficient atio esults in geate sepaation, but if the atio is too high, the desoption time scale must be significantly lengthened to inhibit excessive mass etention of the moe slowly diffusing species in the polyme paticles. As a esult, a high diffusivity atio will also esult in a longe time to each steady-state fo the entie soption/desoption pocess, as it will take moe cycles fo the mass etention to each a pseudo-equilibium. The optimal sepaation paametes ae high diffusivities, a modeate diffusivity atio, high solubilities, and a high solubility atio. 4.3 Specifications fo Cabon Dioxide/Methane Sepaation As discussed peviously, CO 2 and CH 4 have diffeent molecula chaacteistics which allow fo thei sepaation. Two examples of polyimides that best capitalize on these diffeences ae 6FDA-ODA and 6FDA-p-DDS, whose stuctues can be seen in Figue 4.1. The ODA has a highe diffusion coefficient atio, a lowe solubility coefficient atio, highe diffusivities, and lowe solubilities elative to p-dds (Table 4.1) Compaison to Polyme Chaacteistics fo CO 2 /CH 4 Sepaation Polyme compaisons wee conducted on the basis of seveal pocess factos, including soption sepaation, desoption sepaation, and oveall cycle time based on the attainment of an equilibium mass etention inside the polyme paticles. Many of the sepaation pocesses fo CO 2 /CH 4 mentioned in Chapte 1 utilize feed compositions nea a 50/50 mixtue; 3 theefoe, the model calculations completed hee use the same mola 26

34 feed atio. Fo all calculations involving 6FDA-ODA and 6FDA-p-DDS, the desoption time was set equal to thee times the soption time to ensue thee was no significant change in mass etention in the paticles ove long time scales. The soption time fo each polyme was chosen to be the time at which maximum sepaation occued in the extenal phase, based on the extenal concentation atio. Fo long soption times, the extenal concentation atio cuve is not as steep, allowing moe choices fo the optimal sepaation time scale. This can be seen in Figue 4.2 fo CO 2 /CH 4 sepaation. Lastly, the paticle adius fo both polymes was fixed at 50µm. CCH4/CCO2 Figue 4.2. Plot of extenal concentation atio vs. time fo CO 2 /CH 4 sepaation using 6FDA- ODA paticles with a adius of 50µm in the dynamic pessue-swing soption pocess fo a 50/50 feed mixtue. The soption time was detemined to be at the time at which the extenal concentation atio was at a maximum. In the case of 6FDA-ODA, the maximum atio occued at 86s. 27

35 Anothe impotant issue to take into consideation is the attainment of steadystate conditions fo the pocess, because the sepaation chaacteistics ae dependent on the amount of mass etained in the polyme paticles. This has the geatest effect on the desoption composition, seen in Figue 4.3, whee thee is a 3% diffeence in sepaation fom the fist to the last cycle. In ode to detemine the time to each a constant mass uptake fo a given polyme, calculations fo 20 successive cycles wee un. An impotant visual efeence is the shape of the concentation pofiles in the polyme paticles at the end of the desoption step. If the final pofile of a given component has a significant maximum anywhee othe than at the cente of the paticle, the mass build-up of the given component has not eached pseudo-equilibium, and the paticle will continue to etain additional mass. Once the final pofile exhibits an oveall constant o deceasing tend fom the cente of the paticle to the inteface, steady-state has been eached and additional mass will not migate towads the cente of the paticle and accumulate afte successive uns. This behavio can be seen in compaing Figues fo the sepaation of CO 2 and CH 4 using 6FDA-ODA. Based on the model esults fo a 50/50 mixtue, the 6FDA-p-DDS showed only a 1% impovement in sepaation fo the soption step and a 3% impovement in the desoption step (Table 4.2). Both polymes showed nealy the same CH 4 mass etention at the end of the desoption step, but steady-state mass uptake occued in the p-dds deivative afte 12 cycles, wheeas steady-state occued in the ODA deivative in 14 cycles. Plots of the mass uptake fo successive cycles can be seen in Figue 4.8. Lastly, the soption step times wee calculated to be 130s fo the p-dds deivative and 86s fo 28

36 (a) (b) Figue 4.3. Plots of mole faction at the end of the (a) soption and (b) desoption steps fo successive cycles using 6FDA-ODA fo sepaation of a 50/50 mixtue of CO 2 /CH 4 using the dynamic pessue-swing soption pocess. 29

37 (a) time (b) time Figue 4.4. Plots of the concentation pofiles fo the initial soption step in the spheical polyme paticles ove time fo pessue-swing soption sepaation of a 50/50 mixtue of CO 2 /CH 4 using 6FDA-ODA. (a) CO 2 concentation pofiles. (b) CH 4 concentation pofiles. The time steps fo both plots wee t = s. 30

38 (a) Final pofile Initial pofile time (b) Initial pofile Final pofile time Figue 4.5. Plots of concentation pofiles fo the initial desoption step in the polyme paticles ove time fo pessue-swing soption sepaation of a 50/50 CO 2 /CH 4 gas mixtue. (a) CO 2 concentation pofiles. (b) CH 4 concentation pofiles. The initial and final concentation pofiles ae highlighted. The time steps fo each plot wee t = s. 31

39 (a) Initial pofile Final pofile time (b) Final pofile Initial pofile time Figue 4.6. Plots of steady-state soption concentation pofiles in the polyme paticles ove time fo pessue-swing soption sepaation of a 50/50 CO 2 /CH 4 gas mixtue. (a) CO 2 concentation pofiles. (b) CH 4 concentation pofiles. The initial and final concentation pofiles ae highlighted. The time steps fo each plot wee t = s. 32

40 (a) Final pofile Initial pofile time (b) Initial pofile time Final pofile Figue 4.7. Plots of steady-state desoption concentation pofiles in the polyme paticles ove time fo pessue-swing soption sepaation of a 50/50 CO 2 /CH 4 gas mixtue. (a) CO 2 concentation pofiles. (b) CH 4 concentation pofiles. The initial and final concentation pofiles ae highlighted. The time steps fo each plot wee t = s. 33

41 (a) (b) Figue 4.8. Plot of moles of CO 2 and CH 4 emaining in the polyme paticles afte successive sepaation cycles based on a 50/50 feed mixtue at 20 atm, 1L total volume, equal volumes of polyme and extenal phases. (a) 6FDA-ODA. (b) 6FDA-p-DDS. 34

42 Polyme Soption Step (Gas Phase) Desoption Step (Gas Phase) X CO2 /X CH4 Step Time (s) X CO2 /X CH4 Step Time (s) 6FDA-ODA 0.19 / / FDA-p-DDS 0.18 / / Table 4.2. Compaison of polyme pefomance in sepaating a 50/50 mixtue of CO2 and CH4 using pessue swing soption at an initial pessue of 20 atm the ODA deivative (Table 4.2). This step time diffeence esults in a 66% eduction in oveall cycle time fo ODA elative to p-dds. Given the small diffeence in sepaation gained by the p-dds goup compaed to the ODA goup, the bette of the two polymes fo this pocess is 6FDA-ODA given the significantly smalle cycle time length. This eduction in cycle time inceases flow ate and poduction, which allows fo moe efficient opeation of this pocess Compaison to Membane Technology fo CO 2 /CH 4 Sepaation In ode to detemine the elative effectiveness of the pessue-swing pocess, some simple calculations wee made to detemine the sepaation ability of a membane composed of the same polyme fo compaison. The membane calculation used is based on mateial balance equations, tanspot equations, and ideal selectivity, and was taken fom Sepaation Pocess Pinciples. 16 The membane calculation equies pemeabilities as defined by Equation 4.1, as well as a stage cut, defined as: n p θ = (4.2) n f 35

43 Pemeate (n p ) MEMBRANE Feed (n f ) Retentate (n ) Figue 4.9. Schematic of a simple steady-state membane sepaation pocess. whee n p is the mola flow ate of the pemeate and n f is the mola flow ate of the feed. These definitions ae explained in Figue 4.9. The stage cut used fo the membane compaison fo both polysulfone (PSF) and 6FDA-ODA was the same as that fo the feed and moles emoved afte the desoption step fo CO 2 /CH 4 sepaation using 6FDA-ODA. The feed flow ate and pemeate flow ate wee taken as the numbe of moles of feed and moles emoved fom the polyme at the end of the desoption step, espectively, divided by the total cycle time. All nonlinea equations wee solved using Polymath 5.1. Since most commecial membanes have an active laye thickness 17 of Å, a thickness of 1000Å was used fo these calculations. Fo the puposes of the membane compaison calculation, the feed pessue was assumed to be held constant at 20 atm, and the pemeate pessue was assumed to be 0 atm. The membane pemeabilities fo 6FDA-ODA wee found using the data in Table 4.1 and Equation 3.5. The pemeability 11 fo CO 2 in PSF is 36

44 Soption X A4, X B4 Soption X A2, X B2 Sepaation Bed 2 nd Stage Feed X A1, X B1 Sepaation Bed 1 st Stage Desoption X A5, X B5 Soption X A6, X B6 Desoption X A3, X B3 Sepaation Bed 2 nd Stage Desoption X A7, X B7 Figue Simple schematic fo the dynamic pessue-swing soption pocess with a 2 nd stage added cm 3 (STP) cm -1 atm -1 s -1 and the pemeability atio 11 fo CO 2 /CH 4 sepaation is A second stage was added to the pessue-swing soption pocess to compae to membane pefomance. The second stage was calculated by passing the sepaated gas fom the fist soption and desoption steps though anothe set of sepaation beds. This concept is shown in Figue Fo the puposes of the second stage calculation, the mole factions of the soption and desoption steps of the second stage wee used at a pessue of 20 atm. Compaing the membane etentate to the soption step and the pemeate to the desoption step puge in this pocess, a membane composed of 6FDA-ODA gives a 37

45 etentate that is nealy the same as the gas in the extenal phase, and a pemeate that is 17% highe in CO 2 that the extenal phase afte a desoption. If a second stage is applied to the CH 4 -ich steam, a vey high sepaation (95% CH 4 ) is achieved in the soption step fom the second sepaation bed. The CO 2 -ich steam can also be sepaated futhe by a second sepaation bed, esulting in a steam vey close in composition to the membane pemeate. These values ae compaed in Table 4.3, whee all dynamic pessue-swing soption values given ae those calculated fo the pocess afte 20 successive cycles. The 40/60 mixtues fom the desoption step fom the second stage of the CH 4 -ich steam (x A5 /x B5 in Figue 4.10) and fom the soption step of the second stage of the Pocess Pessue-Swing Soption: (6FDA-ODA) Soption Step (Gas Phase) X CO2 / X CH4 Stage Cut (θ) Desoption Step (Gas Phase) X CO2 / X CH4 1 st Stage 0.19 / / nd Stage w/ 1 st Stage Soption Poduct 0.05 / / nd Stage w/ 1 st Stage Desoption Poduct 0.40 / / 0.10 Membane: 6FDA-ODA 0.17 / 0.83 (etentate) / 0.08 (pemeate) PSF 0.14 / 0.86 (etentate) / 0.22 (pemeate) Table 4.3. Results of sepaating a 50/50 mixtue of CO 2 and CH 4 using the pessue-swing soption pocess fo 6FDA-ODA at an initial pessue of 20 atm. The 2 nd stage calculations used the mole factions fom the soption step and desoption step puges at a pessue of 20 atm. These esults ae compaed to a membane pocess. 16 6FDA-ODA and PSF popeties wee used fo the membane calculations. 38

46 CO 2 -ich steam (x A6 /x B6 in Figue 4.10) suggest the potential to ecycle these back to the feed fo geate pocess efficiency. With only one second stage applied to the CO 2 -ich steam fom the fist stage, the pessue-swing soption is compaable to a membane sepaation with the same polyme (Table 4.3). Since PSF is used to make a commecial CO 2 /CH 4 membane, 2 a calculation using its popeties was also completed to compae to the pessue-swing pocess. The single-stage pocess compaes vey well in tems of sepaation pefomance and is within 5% of the etentate composition and within 3% of the pemeate composition. A second stage fo the two gas steams extacted afte the soption and desoption steps shows significant impovement ove the PSF membane, as shown in Table Specifications fo Oxygen/Nitogen Sepaation Since O 2 and N 2 have vey simila chaacteistics, they have histoically been vey difficult to sepaate. Howeve, polyimides have shown pomise in poviding geate sepaation, as compaed to typical membane mateials such as polycabonate and polysulfone. 11 Two such types of polyimides ae 6FDA-DAF and the copolyimide (80/20) 6FDA-duene/mPDA (Figue 4.1). The attached duene and mpda goups povide a highe diffusion coefficient atio, a lowe solubility atio (nea unity), highe diffusivities, and highe solubilities elative to the DAF goup (Table 4.1) Compaison to Polyme Chaacteistics fo O 2 /N 2 Sepaation The same factos fom the CO 2 /CH 4 pocess wee consideed fo the O 2 /N 2 sepaation pocess: sepaation in the extenal phase duing the soption step, oveall cycle 39