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

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

3. Tianjin University 3 Background and goal 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 (ε i ). 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.

4. Tianjin University 4 Stochastic models dX(t, ω) = f(t, X(t, ω)) dt + g(t, X(t, ω)) dW(t, ω) Stochastic models are often derived based on the dynamics of deterministic models. pressure drop conservation, mass conservation stochastic process the phases distribution variation ( q L1, q L2, q G1, q G2 ) Ito SDE: continuous-time Markov chain(CTMC): discrete-time Markov chain(DTMC) : transition probabilities p yx (∆t)=Prob{Y(t+∆t)=y|Y(t)=x}= p(t+∆t)=P p(t)P=(p yx (∆t)), stochastic matrix state changes, probabilities “ forget the past ”

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

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

7. Tianjin University 7 SDE construction 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. complex : nonlinear irregular : random

8. 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 [∆P G (∆t)- ∆P L (∆t)]. Let [X 1 (t), X 2 (t)] T denotes pressure drop at bifurcations, while ∆X 1 (t), ∆X 1 (t) means the pressure drop changes at bifurcations. E.Allen. Modeling with Ito Stochastic Differential Equations[B].2007.

9. Tianjin University 9 SDE construction improvable: all probabilities depend on X 1, X 2 and ∆t patterns + squeneces tend to optimizing and stability

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

11. Tianjin University 11 SDE construction Initial flow distribution ( q L10, q L20, q G10, q G20 ) Pressure change at ∆t Pressure drop and mass conservation Next flow distribution ( q L1, q L2, q G1, q G2 ) Calculate pressure changes through SDE Recursion n times for n∆t Export probability distributions of the solutions, such as E(L bubble ), σ(L bubble ), σ(∆P mix ) etc.

12. Tianjin University 12 Follow-up completion More pressure drop consideration: : Interface renewing of exiting bubbles: Wong et al, for curved caps ： Prove some supposes via SDE models: Whether the gas prior produce the bubble in the channel with the highest gas phase pressure at bifurcations or the lowest pressure drop in the following mix channels. 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. 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.

13. 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.

14. Tianjin University 14 Confusing tips  how to construct random probabilities expressions with deterministic matters(t 1 for liquid slugs and t 2 for bubbles).  how to introduce valuable parameters or fitting parameters.  find a way for flow fluxes recursion.  how to reflect channeling phenomenon from models.

15. 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)