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Collocation Method for the Modeling of Membrane Gas Permeation Systems

Anna Feichtinger, Aleksander Makaruk, Ewa Weinmüller, Anton Friedl and Michael Harasek

Abstract

In this work, we describe a numerical method which enables an efficient computation of membrane gas permeation processes that involve multiple membrane stages and multiple gas components. The utilized numerical approach is a collocation method equipped with a grid adaptation strategy based on a dependable error estimate of the numerical approximation. The comparison of the results provided by the collocation method with those calculated from an experimentally validated finite difference method has demonstrated that the accuracy of both numerical approximations is practically the same. However, the current procedure is characterized by a much better computational efficiency that allows to considerably reduce the computational time. This is a crucial feature when combining computation of membrane permeation processes with optimization algorithms. In such a setting the computation of the permeation process is frequently repeated and naturally, results in long computational times when the efficiency is not adequately improved.

PACS® (2010).: 87.16.dp; 02.60.Lj

Nomenclature

Symbols:
a

left end of the integration interval

A

membrane area

b

right end of the integration interval

B

matrix in the boundary conditions

c(h,z)

error constant

D

diameter of active layer

est

estimate for the global error of the collocation

f

total gas volume flow

f(t,z(t))

right-hand side of the ODE

F

volume flow on the feed side

h

step size

I

subinterval in the grid

J

gas volume flow

k

number of gas components

l

module length

m

number of collocation points

n

dimension of z or f

N

number of subintervals

p

pressure

p

convergence order

P

volume flow on the permeate side

P

piecewise polynomial function

s

number of hollow fibers

S

number of stages

t

longitudinal distance along fiber

TOL

tolerance

x

nonuniform grid

z

solution vector

β

vector in the boundary conditions

Δ

equidistant partition of the interval

Π

proportionality factor (permeance)

χ

volume fraction

τ

endpoints of the grid subintervals

ρ

distribution of the collocation points

ξ

auxiliary uniform grid

ϕ

derivative of Φ

Φ

grid deformation function

Subscripts & superscripts:
0

left integration boundary

1

right integration boundary

F

feed

h

step size

i

gas component

i

vector component

i

subinterval index

j

gas component

k

number of gas components

m

number of collocation points

N

number of subintervals

P

permeate

Δ

equidistant partition of the interval

References

[1] R. W.Baker, Future directions of membrane gas separation technology, Ind. Eng. Chem. Res. 41, 6 (2002), 13931411. Search in Google Scholar

[2] B. D.Bhide and S. A.Stern, A new evaluation of membrane processes for the oxygen-enrichment of air. i. Identification of optimum operating conditions and process configuration, J. Membr. Sci. 62, 1 (1991), 1335. Search in Google Scholar

[3] B. D.Bhide and S. A.Stern, A new evaluation of membrane processes for the oxygen-enrichment of air. ii. Effects of economic parameters and membrane properties, J. Membr. Sci. 62, 1 (1991), 3758. Search in Google Scholar

[4] R. W.Baker and K.Lokhandwala, Natural gas processing with membranes: An overview, Ind. Eng. Chem. Res. 47, 7 (2008), 21092121. Search in Google Scholar

[5] B. D.Bhide and S. A.Stern, Membrane processes for the removal of acid gases from natural gas. i. Process configurations and optimization of operating conditions, J. Membr. Sci. 81, 3 (1993), 209237. Search in Google Scholar

[6] B. D.Bhide and S. A.Stern, Membrane processes for the removal of acid gases from natural gas. ii. Effects of operating conditions, economic parameters, and membrane properties, J. Membr. Sci. 81, 3 (1993), 239252. Search in Google Scholar

[7] H.Lin, S. M.Thompson, A.Serbanescu-Martin, J. G.Wijmans, K. D.Amo, K. A.Lokhandwala and T. C.Merkel, Dehydration of natural gas using membranes. Part i: Composite membranes, J. Membr. Sci. 413–414 (2012), 7081. Search in Google Scholar

[8] H.Sijbesma, K.Nymeijer, R.van Marwijk, R.Heijboer, J.Potreck and M.Wessling, Flue gas dehydration using polymer membranes, J. Membr. Sci. 313, 1–2 (2008), 263276. Search in Google Scholar

[9] X.Feng, S.Sourirajan, F.Handan Tezel, T.Matsuura and B. A.Farnand, Separation of volatile organic compound/nitrogen mixtures by polymeric membranes, Ind. Eng. Chem. Res. 32(3) (1993), 533539. Search in Google Scholar

[10] Y.Liu, X.Feng and D.Lawless, Separation of gasoline vapor from nitrogen by hollow fiber composite membranes for VOC emission control, J. Membr. Sci. 271, 1–2 (2006), 114124. Search in Google Scholar

[11] G.Chatterjee, A. A.Houde and S. A.Stern, Poly(ether urethane) and poly(ether urethane urea) membranes with high h2s/ch4 selectivity, J. Membr. Sci. 135, 1 (1997), 99106. Search in Google Scholar

[12] A.Makaruk, M.Miltner and M.Harasek, Biogas desulfurization and biogas upgrading using a hybrid membrane system – modeling study, Water Sci. Technol. 67, 2 (2013), 326332. Search in Google Scholar

[13] B.Wilks and M. E.Rezac, Properties of rubbery polymers for the recovery of hydrogen sulfide from gasification gases, J. Appl. Polym. Sci. 85, 11 (2002), 24362444. Search in Google Scholar

[14] A.Makaruk, M.Miltner and M.Harasek, Membrane biogas upgrading processes for the production of natural gas substitute, Sep. Purif. Technol. 74, 1 (2010), 8392. Search in Google Scholar

[15] W. J.Schell and C. D.Houston, Use of membranes for biogas treatment, Energy Prog. 3 (1983), 96100. Search in Google Scholar

[16] S. A.Stern, B.Krishnakumar, S. G.Charati, W. S.Amato, A. A.Friedman and D. J.Fuess, Performance of a bench-scale membrane pilot plant for the upgrading of biogas in a wastewater treatment plant, J. Membr. Sci. 151, 1 (1998), 6374. Search in Google Scholar

[17] A.Makaruk, M.Miltner and M.Harasek, Membrane gas permeation in the upgrading of renewable hydrogen from biomass steam gasification gases, Appl. Therm. Eng. 43 (2012), 134140. Search in Google Scholar

[18] T.Mayer, A.Makaruk, N.Diaz, K.Bosch, M.Miltner, M.Harasek and H.Hofbauer. Efficient biomass utilization by polygeneration processes – production of hydrogen, electricity and heat. In Proceedings of ICPS 10 – International Conference on Polygeneration Strategies, Leipzig, 2010. Search in Google Scholar

[19] A.Makaruk and M.Harasek, Numerical algorithm for modelling multicomponent multipermeator systems, J. Membr. Sci. 344, 1–2 (2009), 258265. Search in Google Scholar

[20] S. A.Stern, J. E.Perrin and E. J.Naimon, Recycle and multimembrane permeators for gas separations, J. Membr. Sci. 20, 1 (1984), 2543. Search in Google Scholar

[21] J. G.Wijmans and R. W.Baker, The solution-diffusion model: A review, J. Membr. Sci. 107, 1–2 (1995), 121. Search in Google Scholar

[22] G.Kitzhofer, G.Pulverer, O.Koch, Ch.Simon and E.Weinmüller, The new MATLAB code bvpsuite for the solution of singular implicit bvps, J. Numer. Anal. Indust. Appl. Math5 (2010), 113134. Search in Google Scholar

[23] C.de Boor and B.Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582606. Search in Google Scholar

[24] G.Pulverer, G.Söderlind and E.Weinmüller, Automatic grid control in adaptive BVP solvers, Numer. Algorithms56 (2011), 6192. Search in Google Scholar

[25] O.Koch, R.März, D.Praetorius and E. B.Weinmüller, Collocation methods for index-1 DAEs with a singularity of the first kind, Math. Comp. 79 (2010), 281304. Search in Google Scholar

[26] G.Kitzhofer, O.Koch and E.Weinmüller, Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg–Landau equation, BIT49 (2009), 217245. Search in Google Scholar

Received: 2014-1-3
Accepted: 2015-4-20
Published Online: 2015-5-12
Published in Print: 2015-6-1

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