Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: Primary: 35L65, 65M08; Secondary: 35L45, 35Q79, 92E20, 76T99.

 Citation:

• Figure 1.  Contact discontinuity. Exact and computed solutions at $t = 1$ along the column $0\leq x\leq 0.1$

Figure 2.  Contact discontinuity: $-\log(Erel)$ vs $-\log(\Delta t)$

Figure 3.  Contact discontinuity: CPU time vs $N$

Figure 4.  Solution to the Riemann Problem for $c$

Figure 5.  Adsorption step with the BET isotherm. Exact and computed solutions at $t = 1$ along the column $0\leq x\leq 0.1$

Figure 6.  Function $f$ associated with the binary Langmuir isotherm

Figure 7.  Adsorption step with the binary Langmuir isotherm. Exact and computed solutions at $t = 1$ along the column $0\leq x\leq 0.1$

Figure 8.  Desorption step (bottom) with the binary Langmuir isotherm. Exact and computed solutions at $t = 1$ along the column $0\leq x\leq 0.1$

Figure 9.  Rarefaction: $-\log(Erel)$ vs $-\log(\Delta t)$

Figure 10.  Rarefaction: CPU time vs $N$

Figure 11.  Data for the Riemann problem

Figure 12.  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

•  [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. [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. [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. [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. [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. [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. [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. [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. [9] Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Num. Anal., 21 (1984), 1013-1037.  doi: 10.1137/0721063. [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. [11] C. Dafermos, Hyperbolic Conservation Laws in Continuum physics, Springer, Heidelberg, 2000. doi: 10.1007/3-540-29089-3_14. [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. [13] E. Godlewski and P. A. Raviart, Hyperbolic Systems on Conservation Laws, SMAI, 1991. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] B. Perthame, Kinetic formulation of conservation laws, in Oxford Lecture Series in Mathematics and its Applications, 21, Oxford University Press, Oxford, 2002. [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. [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. [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. [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. [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.

Figures(12)