Wörner, Martin

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KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz AssociationINSTITUTE FOR NUCLEAR AND ENERGY TECHNOLOGIESwww.kit.eduA key parameter to characterize Taylor flow in narrow circular and rectangular channelsDr. Martin WörnerKarlsruhe Institute of TechnologyInstitute for Nuclear and Energy TechnologiesInstitute for Nuclear and Energy Technologies2 Dr. Martin WörnerOutlineIntroductionRelevance of Taylor flowObjectiveProblem descriptionThe key parameter   UB/JtotRelations by definitionBubble diameter and liquid film thicknessRecirculation flow and bypass flowCorrelations for Conclusions03.06.2010Institute for Nuclear and Energy Technologies3Relevance of Taylor flowDr. Martin Wörner03.06.2010Taylor flow is a special kind of slug flow in small channels, where the liquid plugs which separate the elongated bullet-shaped bubbles (Taylor bubbles) are free from gas entrainmentIMM, Mainz (Hessel)Delft group (Moulijn)MIT (Jensen)Institute for Nuclear and Energy Technologies4Advantages of Taylor flowLarge specific interfacial areaefficient heat and mass transferSegmentation of the liquidreduced axial dispersionGood mixing in gas and liquid due to recirculationDr. Martin Wörner03.06.2010Movie from von Günther et al.Langmuir 21 (2005) 1547-1555Channel cross-section:400 µm  280 µmInstitute for Nuclear and Energy Technologies5ObjectiveMany features of Taylor flow such as the liquid film thickness depend on the capillary number CaBThe capillary number is not known a priory for given gas and liquid flow rates, but adjusts accordinglyObjective:propose the use of correlations for global and local quantities in Taylor flow in terms of “a priori” known flow parameterspropose a unified consistent approach, which relates many global and local features of Taylor flow to a single functional dependenceDr. Martin Wörner03.06.2010L BBUCa where UB = bubble velocityInstitute for Nuclear and Energy Technologies6Similitude analysis for a unit cellPhysical properties (6)Gas and liquid density (G, L) and viscosity (G, L)Coefficient of surface tension (), gravity constant (g)Flow specific quantities (3)Gas and liquid flow rates (QG,QL), pressure drop (puc)Geometrical quantities for a rectangular channel (5)Angle of channel orientation () with respect to gChannel height (H) and width (B)Length of unit cell (Luc)Bubble volume in the unit cell (VB)Basic dimensions: kg, m, s (3)Pi-theorem: there are 14 – 3 = 11 independent non-dimensional groupsDr. Martin Wörner03.06.2010unit cellInstitute for Nuclear and Energy Technologies7Pi-theoremA priori unknown groupsA priori known groupsFunctional relationshipsDr. Martin Wörner03.06.2010uch1LD   ch2uBcA LV   ch23uc ucucL G L L ot2tp pEuAQ Q J      L4G   L5G    GG L6QQ Q  G L L totLJch7Q Q JCaA    L h8 2LDLa   L G29h( )g D Eö    10HB   11  u Jc( , , , , , , , , , )Ca La EEF u ö      c Ju ( , , , , , , , , , )Eu Ca La EöEu F        unknown parameterknown parameterInstitute for Nuclear and Energy Technologies8Further non-dimensional groupsSome dependent non-dimensional groupsThe key parameter Dr. Martin Wörner03.06.2010L h totJ JLD JRe LaCa 22L h totJ J J JD JWe Ca Re LaCa  J JtotBB Ca CaJUCa BtotUJ u Jc( , , , , , , , , , )Ca La EöF EuF      Nomenclature   was introduced by Thulasidas et al.(1995)totJ JL GhLJ LaFr CaEögD  Institute for Nuclear and Energy Technologies9Quantities and their relation to Bubble velocityMean liquid velocityGas hold-up in unit cellRelative bubble velocityCapillary numberBubble Reynolds numberDr. Martin Wörner03.06.2010tB otU JtL to11 /U J totBB11UWUJ  tottBot1JJUZ   B JCa CaJ JBR Re Cae La  Institute for Nuclear and Energy Technologies10Liquid mass balance A mass balance for the liquid phase in a frame of reference moving with the bubble at an arbitrary axial position y yieldsso thatDr. Martin Wörner03.06.2010tot B ch L,cs B L( ) ( ) ( )J U A U y U A y    ch toLL,cs t1/( )( )AJyA U yL,csBL,cstotch tot1 /)/( )(( )UJA yAyU yJInstitute for Nuclear and Energy Technologies11Bubble diameter / film thicknessFor a stagnant liquid film the mass balance yieldsFor a circular channel with diameter D it follows for the bubble diameter DB and the liquid film thickness FFor a rectangular channel with an axisymmetric bubble and a stagnant liquid film it followsDr. Martin Wörner03.06.2010hBc1AA B 1DD  F1 112D     B 2DB  B2DH hB 1DDInstitute for Nuclear and Energy Technologies12Recirculation flow and bypass flowDr. Martin Wörner03.06.2010(sketches in moving frame of reference after Taylor 1961),B maxLU UFully developed laminar flow:,max ,mean ( 2; 2.096)L LU CU C C  ,B maxLU UIn Taylor flow it is:,mean totLU J,m aB e n/ LU CU  Condition for bypass flow is,m aB e n/ LU CU  Condition for recirculation flow isInstitute for Nuclear and Energy Technologies13Recirculation area and recirculation timeDr. Martin Wörner03.06.2010For dependence of A0/Ach, A1/Ach and  on  and  in rectangular channels see Kececi, Wörner, Onea, Soyhan, Catalysis Today 147S (2009) S1251 2rR Relations for a circular pipe of radius RThulasidas, Abraham, Cerro, Chem. Eng. Sci. 52 (1997) 29470 22rR0r1rR11 12    Institute for Nuclear and Energy Technologies14Collection of literature dataThulasidas, Abraham, Cerro (1995) (Experiments, circular and square channel)Liu, Vandu, Krishna (2005) (Experiments, circular and square channel)Fairbrother and Stubbs (1935) (Experiments, circular tube)Bretherton (1961) (Theoretical, circular tube)Aussilious and Quere (2000) (Theoretical, circular pipe)Kreutzer, Kapteijn, Moulijn, Heiszwolf (2005) (Square channel)Dr. Martin Wörner03.06.2010J0.33J1for 0.0002 0.391 0.61CaCa   0.5 3 0.5 0.5B totB B JB1for  7.5 10 0.014 1U JCa Ca CaU       2/3 2/3 2/3B totB B JB12.68 for  0.001 1 2.68U JCa Ca CaU    22/3 2/32/3JF BB2/3 2/3 2/3B J1 3.330.66for  0.001 1.41 3.33 1 2CaCaCaD Ca Ca            2B,sq 0.445 0.445 0.445B B Jh40.7 0.5exp 2.25 for 0.04 0.7 0.5exp 2.25DCa Ca CaD          Many relations between and CaJ are implicit in Institute for Nuclear and Energy Technologies15Comparison of literature dataDr. Martin Wörner03.06.2010Liu et al.Fairbrother&StubbsBrethertonAussilious&QuereKreutzer et al.Institute for Nuclear and Energy Technologies16ConclusionsThe ratio  between bubble velocity and total superficial velocity is a key parameter in Taylor flowQuantities that are related to  are the mean liquid velocity, the mean relative velocity, the gas holdupthe local thickness of the liquid film (and the bubble diameter) determines if complete bypass flow or recirculation flow occursThe cross-sectional recirculation area and the non-dimensional recirculation time are unique functions of  (and in rectangular channels of the aspect ratio )Literature data show a clear trend for the dependence of  on CaJIf this functional relation is known, then the hydrodynamics of Taylor flow is almost fully determined  Dr. Martin Wörner03.06.2010uc J J( , , , , , , , , , ) ( , , , , , )F Eu Ca La Eö F Ca La Eö              

A key parameter to characterize Taylor flow in narrow circular and rectangular channels

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A key parameter to characterize Taylor flow in narrow circular and rectangular channels

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