Stochastic models for microchannels numbering-up

effect description

Reporter : Lexiang Zhang

Supervisor : Feng Xin

2012.09.25

Tianjin University2

stochastic and deterministic models

background and goal

SDE construction

confusing tips

perspectives

Contents

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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 (εi).

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.

dpdt = Qp

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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

( qL1 , qL2 , qG1 , qG2 )

Ito SDE:

continuous-time Markov chain(CTMC):

discrete-time Markov chain(DTMC):

transition probabilities pyx (∆t)=Prob{Y(t+∆t)=y|Y(t)=x}=

p(t+∆t)=P p(t) P=(pyx (∆t)), stochastic matrix

state changes, probabilities

“ forget the past ”

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Deterministic model

Voikert et al Proposed :

The pressure drop caused by friction is only taken into account initially,

∆P = 12μuLh2 1− 192hπ5w thቀπw2hቁ൨−1 = KLq= K∆tuq

Vfillhw2 = 3π8 ቀ1−π4ቁhw

t1 = VfillqG

t2 = 1f − VfillqG

The generation frequency(f) partition :

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Deterministic model

∆Pi = ቈKGuGijቆ1fj − VfillqGijቇqGij + KLuLij VfillqGij qLijnj=m

Ld + σ LBni=mLd + σ LB′qi=p = qG1qG2

qG1 + qG2 = qGtot qL1 + qL2 = qLtot

When bubbles(liquid slugs) enter in mix channel, they move with the same velocity, the fluxes differences are reflected on the slugs sizes.

Ld + σ LSni=mLd + σ LS′qi=p = qL1qL2

fixed mix channel volume

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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

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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.

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SDE constructioni State change (∆X)T

i Probability 1 ൫KGuGtot ∆tqGtot − KLuL1∆tqL1 ,0൯ qGtotqGtot +qL1

2 ൫0 ,KGuGtot ∆tqGtot − KLuL2∆tqL2൯ qGtotqGtot +qL2

3 ൫KGuG1∆tqG1 − KLuL1∆tqL1 ,KGuG2∆tqG2− KLuL2∆tqL2൯

qG1qG1 +qL1qG2qG2 + qL2

4 ሺ0 ,0ሻ 1− pi3i=1

i State change (∆X)T

i Probability 1 ൫KGuGtot∆tqGtot− KLuL1∆tqL1 ,0൯ 1f1 − VfillqGtot1f1

2 ൫0 ,KGuGtot ∆tqGtot −KLuL2∆tqL2൯ 1f2 − VfillqGtot1f2

3 ൫KGuG1∆tqG1− KLuL1∆tqL1 ,KGuG2∆tqG2− KLuL2∆tqL2൯ 1f1 −VfillqG11f11f2 − VfillqG21f2

4 ሺ0 ,0ሻ 1− pi3i=1

improvable:all probabilities depend on X1, X2 and ∆t

patterns + squeneces

tend to optimizing and stability

∆Pmix,i = KG∆t uGijqGijn

j=m + Xijn

j=m

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SDE construction

Xn+1 = Xn + 12b∆Wn + 12bሺtn + ∆ ,Xn + a∆+ b∆Wnሻ∆Wn + 12a∆+ 12aሺtn + ∆ ,Xn + a∆+b∆Wnሻ∆− 12b∂b∂x∆ ∆= (T− t0) NΤ

two-stage Runge-Kutta schemes:

a(t,X(t,ω)) = pjj ∆Xj ∆t൘ b(t,X(t,ω)) = ඩ pjj ∆Xj൫∆Xj൯T ∆t൘

dX(t, ω) = a(t, X(t, ω)) dt + b(t, X(t, ω)) dW(t, ω)

numbering-up effect description:

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SDE construction

Initial flow distribution ( qL10 , qL20 , qG10 , qG20 )

Pressure change at ∆t

Pressure drop and mass conservation

Next flow distribution ( qL1 , qL2 , qG1 , qG2 )

Calculate pressure changes through SDE

Recursion n times for n∆t

Export probability distributions of the solutions, such as E(Lbubble), σ(Lbubble), σ(∆Pmix) etc.

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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.

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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.

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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.

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Perspectives

Compete stochastic models and programme for the numerical solutions(matlab)

Plan experiment schemes(relative variation from optical measurment shows advantage from CCD)