Stochastic models for microchannels numbering-up effect description

1. Stochastic models for microchannels numbering-up effect description Reporter : Lexiang Zhang Supervisor : Feng Xin 2012.09.25

2. Contents background and goal stochastic and deterministic models SDE construction confusing tips perspectives Tianjin University

3. Background and goal Almost studies investigated the design methodology in order to get optimum performances, while the micoreactors can put into practice with the acceptable operation deviation. The key point for describing the numbering-up effect among parallel microchannels is two phases flow distribution, which can be reflected from pressure changes at bifurcations. Such an equation mirrors the interaction between bifurcations, the two phases flow distribution, the feedback and crosstalk as well as the channel structure in parallel microchannels, also can predict the channels performances (εi). Tianjin University

4. Stochastic models the phases distribution variation ( qL1 , qL2 , qG1 , qG2 ) pressure drop conservation, mass conservation stochastic process Stochastic models are often derived based on the dynamics of deterministic models. Ito SDE: dX(t, ω) = f(t, X(t, ω)) dt + g(t, X(t, ω)) dW(t, ω) discrete-time Markov chain(DTMC): state changes, probabilities transition probabilities pyx (∆t)=Prob{Y(t+∆t)=y|Y(t)=x}= p(t+∆t)=P p(t) P=(pyx (∆t)), stochastic matrix continuous-time Markov chain(CTMC): “ forget the past ” Tianjin University

5. Deterministic model Voikert et al Proposed : The generation frequency(f) partition : The pressure drop caused by friction is only taken into account initially, Tianjin University

6. Deterministic model fixed mix channel volume When bubbles(liquid slugs) enter in mix channel, they move with the same velocity, the fluxes differences are reflected on the slugs sizes. Tianjin University

7. SDE construction complex : nonlinear irregular : random Two thoughts for SDE construction: • set springboard on bubble formation steps • Such an equation mirrors the interaction between bifurcations, the two phases flow distribution, the feedback and crosstalk. • construct state changes and probabilities from the statistics viewpoint using a mass of experimental data(slug sizes, velocities etc.) rather than objective law. Tianjin University

8. SDE construction The channels are filled with liquid and only consider liquid frictional pressure drop first, when gas enters in time interval ∆t, the pressure drop changes [∆PG(∆t)- ∆PL(∆t)]. Let [X1(t) , X2(t)]T denotes pressure drop at bifurcations, while ∆X1(t), ∆X1(t) means the pressure drop changes at bifurcations. E.Allen. Modeling with Ito Stochastic Differential Equations[B].2007. Tianjin University

9. SDE construction patterns + squeneces tend to optimizing and stability improvable: all probabilities depend on X1, X2 and ∆t Tianjin University

10. SDE construction dX(t, ω) = a(t, X(t, ω)) dt + b(t, X(t, ω)) dW(t, ω) numbering-up effect description: two-stage Runge-Kutta schemes: Tianjin University

11. SDE construction Initial flow distribution ( qL10 , qL20 , qG10 , qG20 ) Pressure change at ∆t Pressure drop and mass conservation Calculate pressure changes through SDE Next flow distribution ( qL1 , qL2 , qG1 , qG2 ) Recursion n times for n∆t Export probability distributions of the solutions, such as E(Lbubble), σ(Lbubble), σ(∆Pmix) etc. Tianjin University

12. : Follow-up completion More pressure drop consideration: Wong et al, for curved caps： R. Sh. Abiev.Modeling of Pressure Losses for the Slug Flow of a Gas–Liquid Mixture in Mini- and Microchannels[J]. Theoretical Foundations of Chemical Engineering.2011,45(2):156-163. Interface renewing of exiting bubbles: M.J.F. Warnier, E.V. Rebrov, M.H.J.M. de Croon et al.Gas hold-up and liquid film thickness in Taylor flow in rectangular microchannels[J]. Chemical Engineering Journal.2008,135:153-158. Prove some supposes via SDE models: Whether the gas priorproduce the bubble in the channel with the highest gas phase pressure at bifurcations or the lowest pressure drop in the following mix channels. Tianjin University

13. SDE construction focus on the pressure changes at bifurcations and take less consideration on pressure drop along mix channels Suppose two phases flow fluxes keep constant during a slug formation. Record the slug lengths, then get a distribution(X axis: slug length; Y axis: occurance), construct SDE on these data. Adam R. Abate,Pascaline Mary, Pascaline Mary et al.Experimental validation of plugging during drop formation in a T-junction[J]. Lab on a chip.2012,2(12):1516-1521. Tianjin University

14. Confusing tips • how to construct random probabilities expressions with deterministic matters(t1 for liquid slugs and t2 for bubbles). • how to introduce valuable parameters or fitting parameters. • find a way for flow fluxes recursion. • how to reflect channeling phenomenon from models. Tianjin University

15. Perspectives • Compete stochastic models and programme for the numerical solutions(matlab) • Plan experiment schemes(relative variation from optical measurment shows advantage from CCD) Tianjin University

16. Thank you!