Blow up at the hyperbolic boundary for a 2x2 system arising from - PowerPoint PPT Presentation

Blow up at the hyperbolic boundary for a 2x2 system arising from chemical engineering Christian Bourdarias, Marguerite GISCLON, Stéphane Junca University of Savoie, LAMA, UMR-CNRS-5127 Université de Nice Sophia-Antipolis, Labo JAD & Epi COFFEE INRIA 26 Juin 2012 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 1 / 38

Contents Introduction 1 Chromatography The model Sorption effect Gas-chromatography with sorption effect for two species 2 Hyperbolicity, Riemann invariants Entropies Riemann Problem Existence of entropy solutions 3 Godunov scheme Front Tracking Algorithm (FTA) An example of “Blow up” 4 Prospects and open problems 5 Bibliography 6 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 2 / 38

Contents Introduction 1 Chromatography The model Sorption effect Gas-chromatography with sorption effect for two species 2 Hyperbolicity, Riemann invariants Entropies Riemann Problem Existence of entropy solutions 3 Godunov scheme Front Tracking Algorithm (FTA) An example of “Blow up” 4 Prospects and open problems 5 Bibliography 6 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 3 / 38

What is it ? Chromatography gas-solid : a technic for analyzing a gaseous mixture The mixture analysed is vaporised at the entrance of a column that contains a solid substance (the absorber) called the stationary phase and then it is transported across it by a gas carrier. The gas carrier (or gas vector) is the mobile phase. In most cases it has to be inerted vis a vis the solutes and the stationary phase. (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 4 / 38

Unknowns During the processus, d species exist together under 2 phases one gas phase of concentration c i and moving with speed u one solid (adsorbed) with concentration q ∗ i We define by u the speed of a tracer that is inert in the column. This is a measure of the speed of the flow across the column. In general this speed can not be considered constant. (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 5 / 38

Non linear chromatography Chemical assumptions constant temperature 1D model, x ∈ I R + negligeable axial diffusion perfect gas instantaneous exchange Prospects nonisothermal model validity of the model relaxation (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 6 / 38

Equations Mass conservation : ∂ t ( c i + q ∗ i ) + ∂ x ( u c i ) = 0 , 1 ≤ i ≤ d q ∗ i depends on all c i : q ∗ i = q ∗ i ( c 1 , · · · , c d ) i is the i th isothermal law given by the experiments (chemistery) q ∗ (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 7 / 38

Classic isotherms Linear isotherm : q ∗ i = K i c i , avec K i ≥ 0 Langmuir isotherm : Q i K i c i q ∗ i = , with K i ≥ 0 , Q i > 0 d � 1 + K j c j j = 1 BET isotherm : one active gas and one inert gas ( q ∗ 2 = 0) Q K c 1 q ∗ 1 = ( 1 + K c 1 − ( c 1 / c s ))( 1 − ( c 1 / c s )) , Q > 0 , K > 0 , c s > 0 , (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 8 / 38

Sorption effect Unknowns : velocity u , gazeous concentrations c i In gas-solid chromatography, the velocity u is not constant : it depends on the mixture composition, which depends on mass transfer between phases. The velocity of the mixture has to be found in order to achieve a given pressure (or density in the isothermal model). The sorption effect is taken into account through a constraint d � c i = ρ ( t ) i = 1 ρ : the given total density of the mixture (the same in the column) (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 9 / 38

Contents Introduction 1 Chromatography The model Sorption effect Gas-chromatography with sorption effect for two species 2 Hyperbolicity, Riemann invariants Entropies Riemann Problem Existence of entropy solutions 3 Godunov scheme Front Tracking Algorithm (FTA) An example of “Blow up” 4 Prospects and open problems 5 Bibliography 6 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 10 / 38

d = 2 ρ ≡ 1 and x ∈ I R + : ∂ t ( c 1 + q ∗ 1 ( c 1 , c 2 )) + ∂ x ( u c 1 ) = 0 (1) ∂ t ( c 2 + q ∗ 2 ( c 1 , c 2 )) + ∂ x ( u c 2 ) = 0 (2) c 1 + c 2 = 1 (3) Let c = c 1 and q i ( c ) = q ∗ i ( c , 1 − c ) i = 1 , 2 (1) + (2) yields with (3) : ∂ t ( q 1 ( c ) + q 2 ( c )) + ∂ x u = 0 1 = dq 1 2 = dq 2 N.B. : isotherm properties : q ′ dc ≥ 0 and q ′ dc ≤ 0 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 11 / 38

The 2 × 2 system  ∂ t ( c + q 1 ( c )) + ∂ x ( u c ) = 0      ∂ t ( q 1 ( c ) + q 2 ( c ) � ) + ∂ x u = 0  � ��     = h ( c ) Initial boundary values :  c ( 0 , x ) = c 0 ( x ) ∈ [ 0 , 1 ] , x > 0    c ( t , 0 ) = c b ( t ) ∈ [ 0 , 1 ] , t > 0    u ( t , 0 ) = u b ( t ) > 0 , t > 0 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 12 / 38

The 2 × 2 system We analyse the system in terms of hyperbolicity system of P .D.E provided we exchange the time and space variables : x is the evolutive variable, not t � ∂ x ( u c ) + ∂ t ( c + q 1 ( c )) = 0 ∂ x u + ∂ t h ( c ) = 0 with conservative quantities m = u c , u = u ρ m is the flow rate of the first species u ρ is the total flow rate (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 13 / 38

Hyperbolicity Eigenvalues : 0 and λ = H ( c ) u with H ( c ) = 1 + ( 1 − c ) q ′ 1 − c q ′ 2 ≥ 1 If u > 0, the system is strictly hyperbolic. Moreover : d λ · r = H ( c ) f ′′ ( c ) u 2 with r the associated eigenvector to λ and f ( c ) = q 1 ( c ) − c h ( c ) = ( 1 − c ) q 1 ( c ) − cq 2 ( c ) λ is genuinely nonlinear if f ” � = 0 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 14 / 38

Examples Langmuir isotherm : Q 1 K 1 c Q 2 K 2 ( 1 − c ) q 1 ( c ) = 1 + K 1 c + K 2 ( 1 − c ) , q 2 ( c ) = 1 + K 1 c + K 2 ( 1 − c ) f ′′ keeps a constant sign BET isotherm : with an inert gas ( q 2 ( c ) = 0) and Q K c q 1 ( c ) = ( 1 + Kc − ( c / c s ))( 1 − ( c / c s )) , Q > 0 , K > 0 , c s > 0 , then f ( c ) = ( 1 − c ) q 1 ( c ) is not concave and not convex (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 15 / 38

Remarks Generally, isotherms are not convex except some important cases (Langmuir, ammoniac, water vapour) Interpretation : at each inflexion point a new layer starts on the pervious layer (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 16 / 38

Riemann Invariants For smooth solutions ∂ x ( u G ( c )) = 0 ∂ x c + H ( c ) ∂ t c = 0 u with g ′ = − h ′ H , G = exp ( g ) There are two Riemann invariants : W = u G ( c ) = u e g ( c ) c and (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 17 / 38

Entropies Entropies : S ( c , u ) = φ ( uG ( c )) + u ψ ( c ) where φ and ψ are smooth functions Entropy-flux Q = Q ( c ) satisfies Q ′ ( c ) = h ′ ( c ) ψ ( c ) + H ( c ) ψ ′ ( c ) ⇒ ∂ t S + ∂ x Q = 0 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 18 / 38

Convex entropy ? For any convex function ψ , S 2 ( c , u ) = u ψ ( c ) is convex (but not strictly convex) There are strictly convex entropies of the form S 1 ( c , u ) = φ ( uG ( c )) ⇔ G ′′ = ( g ′′ + g ′ 2 ) exp ( g ) � = 0 for c ∈ [ 0 , 1 ] (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 19 / 38

Weak entropy solutions Definition Let T > 0, X > 0, u ∈ L ∞ (( 0 , T ) × ( 0 , X )) and 0 ≤ c ( t , x ) ≤ 1 p.p. in ( 0 , T ) × ( 0 , X ) then ( c , u ) is a weak entropy solution if for any convex function ψ : ∂ x ( u ψ ( c )) + ∂ t Q ( c ) ≤ 0 with Q ′ = H ψ ′ + h ′ ψ Remark : if ± G ′′ > 0 on [ 0 , 1 ] , we also have : ± ∂ x ( u G ( c )) ≤ 0 , on [ 0 , 1 ] (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 20 / 38

Riemann problem Rappel : c is a Riemann invariant If f is convex : through a λ -wave TV [ ln ( u ( z ))] ≤ γ | c + − c 0 | , z = x t (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 21 / 38

Contents Introduction 1 Chromatography The model Sorption effect Gas-chromatography with sorption effect for two species 2 Hyperbolicity, Riemann invariants Entropies Riemann Problem Existence of entropy solutions 3 Godunov scheme Front Tracking Algorithm (FTA) An example of “Blow up” 4 Prospects and open problems 5 Bibliography 6 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 22 / 38

Godunov scheme The convergence of the Godunov scheme is generally an open problem for a 2 × 2 system LeVeque, Randall J. ; Temple, Blake, 1985 Stability of Godunov’s method for a class of 2 × 2 systems of conservation laws Trans. Amer. Math. Soc. 288 Bressan, Alberto ; Jenssen, Helge Kristian, 2001 Convergence of the Godunov scheme for straight line systems Bressan, Alberto ; Jenssen, Helge Kristian ; Baiti, Paolo, 2006 An instability of the Godunov scheme Comm. Pure Appl. Math. 59 (HYP2012) Blow-up for a 2 × 2 hyperbolic system 26 Juin 2012 23 / 38