# American Institute of Mathematical Sciences

October  2020, 13(5): 869-888. doi: 10.3934/krm.2020030

## Kinetic formulation of a 2 × 2 hyperbolic system arising in gas chromatography

 1 Université Savoie Mont Blanc, LAMA, UMR CNRS 5127, 73376 Le Bourget-du-Lac Cedex, France 2 Université Côte d'Azur, LJAD, Inria & CNRS 7351, Parc Valrose, 06108 Nice

Received  May 2019 Revised  March 2020 Published  August 2020

A particular 2x2 hyperbolic system commonly used in the context of gas-solid chromatography is reformulated as a single kinetic equation using an additional kinetic variable. A kinetic numerical scheme is built from this new formulation and its behavior is tested on solving the Riemann problem in different configurations leading to single or composite waves.

Citation: Christian Bourdarias, Marguerite Gisclon, Stéphane Junca. Kinetic formulation of a 2 × 2 hyperbolic system arising in gas chromatography. Kinetic & Related Models, 2020, 13 (5) : 869-888. doi: 10.3934/krm.2020030
##### References:
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Junca, Existence of weak entropy solutions for gas chromatography system with one or two actives species and non convex isotherms, Commun. Math. Sci., 5 (2007), 67-84.  doi: 10.4310/CMS.2007.v5.n1.a3.  Google Scholar [6] C. Bourdarias, M. Gisclon and S. Junca, Blow up at the hyperbolic boundary for a $2 \times 2$ system arising from chemical engineering, J. Hyperbolic Differ. Equ., 7 (2010), 297-316.  doi: 10.1142/S0219891610002116.  Google Scholar [7] C. Bourdarias, M. Gisclon and S. Junca, Strong stability with respect to weak limits for a hyperbolic system arising from gas chromatography, Methods Appl. Anal., 17 (2010), 301-330.  doi: 10.4310/MAA.2010.v17.n3.a5.  Google Scholar [8] C. Bourdarias, M. Gisclon and S. Junca, Eulerian and lagrangian formulations in $BV^{s}$ for gas-solid chromatography, Com. in Math. Sci., 14 (2016), 665-1685.  doi: 10.4310/CMS.2016.v14.n6.a10.  Google Scholar [9] Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Num. 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Phys., 163 (1994), 415-431.  doi: 10.1007/BF02102014.  Google Scholar [20] T. P. Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl., 53 (1976), 78-88.  doi: 10.1016/0022-247X(76)90146-3.  Google Scholar [21] V. I. Oseledets, A new form of writing out the Navier-Stokes equation. The Hamiltonian formalism, Russian Math. Surveys, 44 (1989), 210-212.  doi: 10.1070/RM1989v044n03ABEH002122.  Google Scholar [22] Y.-J. Peng, F. James and B. Perthame, A kinetic formulation for chromatography, in Hyperbolic Problems: Theory, Numerics, Applications (Stony Brook, NY, 1994), World Scientific Pub Co Inc., River Edge, NJ, 1996,354–360.  Google Scholar [23] B. Perthame, Kinetic formulation of conservation laws, in Oxford Lecture Series in Mathematics and its Applications, 21, Oxford University Press, Oxford, 2002.  Google Scholar [24] B. Perthame and A.-E. Tzavaras, Kinetic formulation for systems of two conservation laws and elastodynamics, Arch. Ration. Mech. Anal., 155 (2000), 1-48.  doi: 10.1007/s002050000109.  Google Scholar [25] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), 201-231.  doi: 10.1007/s10092-001-8181-3.  Google Scholar [26] P. Rouchon, M. Sghoener, P. Valentin and G. Guiochon, Numerical simulation of band propagation in nonlinear chromatography, in Chromatographic Science Series, 46, Eli Grushka, Marcel Dekker Inc., New York, 1988. doi: 10.1080/01496398708057614.  Google Scholar [27] L. H. Shendalman and J. E. Mitchell, A study of heatless adsorption in the model system co$_2$ in he, i., Chemical Engineering Science, 27 (1972), 1449-1458.   Google Scholar [28] Henry William, Experiments on the quantity of gases absorbed by water, at different temperatures, and under different pressures, Philosophical Transansactions of the Royal Society of London, 93 (1803), 29-274.   Google Scholar

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##### References:
 [1] E. I. Akulinin, O. O. Golubyatnikov, D. S. Dvoretsky and S. I. Dvoretsky, Numerical study of cyclic adsorption processes of air oxygen enrichment in dynamics, Journal of Physics: Conference Series, (2019), 012005. doi: 10.1088/1742-6596/1278/1/012005.  Google Scholar [2] C. Bourdarias, On a system of p.d.e. modelling heatless adsorption of a gaseous mixture, M2AN, 26 (1992), 867-892.  doi: 10.1051/m2an/1992260708671.  Google Scholar [3] C. Bourdarias, Approximation of the solution to a system modeling heatless adsorption of gases, SIAM J. Numer. Anal., 35 (1998), 13-30.  doi: 10.1137/S0036142993248249.  Google Scholar [4] C. Bourdarias, M. Gisclon and S. Junca, Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases, J. Math. Anal. Appl., 313 (2006), 551-571.  doi: 10.1016/j.jmaa.2005.07.082.  Google Scholar [5] C. Bourdarias, M. Gisclon and S. Junca, Existence of weak entropy solutions for gas chromatography system with one or two actives species and non convex isotherms, Commun. Math. Sci., 5 (2007), 67-84.  doi: 10.4310/CMS.2007.v5.n1.a3.  Google Scholar [6] C. Bourdarias, M. Gisclon and S. Junca, Blow up at the hyperbolic boundary for a $2 \times 2$ system arising from chemical engineering, J. Hyperbolic Differ. Equ., 7 (2010), 297-316.  doi: 10.1142/S0219891610002116.  Google Scholar [7] C. Bourdarias, M. Gisclon and S. Junca, Strong stability with respect to weak limits for a hyperbolic system arising from gas chromatography, Methods Appl. Anal., 17 (2010), 301-330.  doi: 10.4310/MAA.2010.v17.n3.a5.  Google Scholar [8] C. Bourdarias, M. Gisclon and S. Junca, Eulerian and lagrangian formulations in $BV^{s}$ for gas-solid chromatography, Com. in Math. Sci., 14 (2016), 665-1685.  doi: 10.4310/CMS.2016.v14.n6.a10.  Google Scholar [9] Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Num. Anal., 21 (1984), 1013-1037.  doi: 10.1137/0721063.  Google Scholar [10] S. Brunauer, P. H. Emmett and E. Teller, Adsorption of gases in multimolecular layers, Journal of the American Chemical Society, 60 (1938), 309-319.  doi: 10.1021/ja01269a023.  Google Scholar [11] C. Dafermos, Hyperbolic Conservation Laws in Continuum physics, Springer, Heidelberg, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar [12] S. J. Doong and R. T. Yang, Bulk separation of multicomponent gas mixture by pressure swing adsorption: Pore/surface diffusion and equilibrium models, AIChE Journal, 32 (1986), 397-410.  doi: 10.1002/aic.690320306.  Google Scholar [13] E. Godlewski and P. A. Raviart, Hyperbolic Systems on Conservation Laws, SMAI, 1991.  Google Scholar [14] E. Godlewski and P. A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, 118, Springer-Verlag, New-York, 1996. doi: 10.1007/978-1-4612-0713-9.  Google Scholar [15] F. James, Y.-J. Peng and B. Perthame, Kinetic formulation for chromatography and some other hyperbolic systems, J. Math. Pures Appl., 74 (1995), 367-385.   Google Scholar [16] Irving Langmuir, The adsorption of gases on plane surface of glass, mica and platinum, Journal of the American Chemical Society, 40 (1918), 1361-1402.  doi: 10.1021/ja02242a004.  Google Scholar [17] M. Douglas Levan, C. A. Costa, A. E. Rodrigues, A. Bossy and D. Tondeur, Fixed–bed adsorption of gases: Effect of velocity variations on transition types, AIChE Journal, 34 (1988), 996-1005.  doi: 10.1002/aic.690340612.  Google Scholar [18] P.-L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related questions, J. Amer. Math. Soc., 7 (1994), 169-191.  doi: 10.1090/S0894-0347-1994-1201239-3.  Google Scholar [19] P.-L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Comm. Math. Phys., 163 (1994), 415-431.  doi: 10.1007/BF02102014.  Google Scholar [20] T. P. Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl., 53 (1976), 78-88.  doi: 10.1016/0022-247X(76)90146-3.  Google Scholar [21] V. I. Oseledets, A new form of writing out the Navier-Stokes equation. The Hamiltonian formalism, Russian Math. Surveys, 44 (1989), 210-212.  doi: 10.1070/RM1989v044n03ABEH002122.  Google Scholar [22] Y.-J. Peng, F. James and B. Perthame, A kinetic formulation for chromatography, in Hyperbolic Problems: Theory, Numerics, Applications (Stony Brook, NY, 1994), World Scientific Pub Co Inc., River Edge, NJ, 1996,354–360.  Google Scholar [23] B. Perthame, Kinetic formulation of conservation laws, in Oxford Lecture Series in Mathematics and its Applications, 21, Oxford University Press, Oxford, 2002.  Google Scholar [24] B. Perthame and A.-E. Tzavaras, Kinetic formulation for systems of two conservation laws and elastodynamics, Arch. Ration. Mech. Anal., 155 (2000), 1-48.  doi: 10.1007/s002050000109.  Google Scholar [25] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), 201-231.  doi: 10.1007/s10092-001-8181-3.  Google Scholar [26] P. Rouchon, M. Sghoener, P. Valentin and G. Guiochon, Numerical simulation of band propagation in nonlinear chromatography, in Chromatographic Science Series, 46, Eli Grushka, Marcel Dekker Inc., New York, 1988. doi: 10.1080/01496398708057614.  Google Scholar [27] L. H. Shendalman and J. E. Mitchell, A study of heatless adsorption in the model system co$_2$ in he, i., Chemical Engineering Science, 27 (1972), 1449-1458.   Google Scholar [28] Henry William, Experiments on the quantity of gases absorbed by water, at different temperatures, and under different pressures, Philosophical Transansactions of the Royal Society of London, 93 (1803), 29-274.   Google Scholar
Contact discontinuity. Exact and computed solutions at $t = 1$ along the column $0\leq x\leq 0.1$
Contact discontinuity: $-\log(Erel)$ vs $-\log(\Delta t)$
Contact discontinuity: CPU time vs $N$
Solution to the Riemann Problem for $c$
Adsorption step with the BET isotherm. Exact and computed solutions at $t = 1$ along the column $0\leq x\leq 0.1$
Function $f$ associated with the binary Langmuir isotherm
Adsorption step with the binary Langmuir isotherm. Exact and computed solutions at $t = 1$ along the column $0\leq x\leq 0.1$
Desorption step (bottom) with the binary Langmuir isotherm. Exact and computed solutions at $t = 1$ along the column $0\leq x\leq 0.1$
Rarefaction: $-\log(Erel)$ vs $-\log(\Delta t)$
Rarefaction: CPU time vs $N$
Data for the Riemann problem
Shocks chords are shown as dashed lines. On the left $c^-$ is connected to $c^+$ via a shock (S), a rarefaction wave (R) and a shock. On the right, $c^-$ is connected to $c^+$ via a shock and a rarefaction wave
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