`Department of Chemical Engineering Fluids 4 ­ Two Phase FlowTwo phase flow is something that you have met before in your course, for instance in the design of distillation columns and liquid extraction equipment. Other familiar examples include fog, snow, boiling liquids, coffee percolaters, pouring liquid from a bottle or can, etc. In the chemical industry, further examples include fluidised beds, pumping of slurries, pumping of flashing liquids, raining bed driers, e.t.c., and in the oil industry two phase flow occurs in pipelines carrying oil and natural gas. The laws governing two phase flow are identical to those for single phase flow. However, the equations are more complex and/or more numerous than those of single phase flow. In today's lecture I shall describe the types of behaviour found in liquid gas flows and shall say something of the methods by which we analyse such flows. In our next lecture, we shall look further at the analysis techniques to show how pressure drops can be predicted using experimental correlations based on simple analytical models. If we have time we may mention some problems associated with solid/fluid flows. The diagram below, reproduced from &quot;One-Dimensional Two-Phase Flow&quot; by G B Wallis, shows a plot of the typical flow regimes for vertical gas-liquid flows in a pipe.jG m/s10de-wetting small ripplesAnnular Churn1ImpossibleFloodingSlugjF m/s Bubble-1Annular0.1 -0.1Bubble0.01 0.1 1.0jF m/s-1Annular-10Impossiblede-wetting small ripplesjG m/sMap of Flow Regimes: Air-water at STP flowing in a vertical, 2.5cm diameter tube Upward velocities are positiveLet us look at each section of the flow in turn. We shall use the notation jG and jF to denote the superficial velocity of the gas and liquid respectively.Co-current Vertical Upwards Flow1) At low jG we have Bubble Flow jG =QG AjGjF =QF ABubble size depends on nature of air injection. Bubble rise velocity depends on bubble size. If too many bubbles, they coalesce giving large bubbles ­ less coalescence in dirty water. Upper limit to void fraction  10% i.e. Void fraction  = jG = QG  0.1 jF QF 2) Slug Flow ­ large bubbles that almost occupy the whole cross section. Slug rise velocity depends on tube diameter. Rise velocity is independent of viscosity.jF3) At jG + jF = 1.7m/s slug flow becomes unstable giving way to churn flow ­ this is chaotic ­ slugs form and immediately break up ­ not amenable to theoretical analysis.4) Annular Flow For jG  10m/s churn flow becomes gradually steady giving a liquid film on the wall and a central gas core. Problem of waves forming on the liquid surface a) as ripples having wavelength of the same order as film thicknessjGjFor b)Disturbance waves (Kelvin-Helmholtz waves) with amplitude 10 times the film thickness and wavelength 1000 times the film thickness. The velocity of the wave &gt;&gt; film velocity. Droplets are entrained (as much as half of the liquid is entrained at high jG ). 5) Eventually we have Mist FlowjFj jFG­ fine droplets of liquid dispersed in the gas phase ­ some liquid still remains on the wall.jGCountercurrent Vertical FlowOnly liquid flow down and gas up is possible, e.g. wetted wall column ­ annular flow. Can also have bubble flow. At high jG and/or jF flooding occurs. In co-current downward flow the annular flow regime is most common.Horizontal FlowDispersed Bubble 3jF1PlugSlugm/s 0.2 0.1 Wave Stratified Annular Mist0.01 0.1 1.0 4 10 jG100 m/sAgain there are a range of flow regimes dependant on jG and jf 1) Dispersed Bubble FlowjFjG2) Plug Flow ­ elongated bubble3) Slug Flow4) Stratified Flowgasliquid5) Wave Flowgasliquid6) Annlar MistjGjFAnalysis of two-phase flowsHow do we analyse two-phase flows? Let us consider 1-Dimensional analysis techniques. These divide into the following categories. a) Simple Correlations ­ based on experiment ­ often quoted in dimensionless form ­ may or may not have scientific/physical basis ­ often restricted in area of application b) Simple Analytical Models 1) Homogeneous model  take average of properties for both phases  used, e.g., for suspension, foam, mist, dispersed bubble  no detail of the flow considered  assume phases flow side by side  use separate equations for each phase  consider interation between the phases  focuses on relative motion between phases c) Integral Analysis ­ assume velocity, temperature or concentration profile ­ fit to boundary conditions and apply integrated fluid mechanical equations d) Differential Analysis ­ use of time-averaged equations of motion e) Universal Phenomena ­ certain phenomena apply regardless of the regime, e.g. wave theory2) Separated flow model3) Drift flux modelTwo Phase Flow Dimensionless GroupsNB  = F - G which is often  F Drag coefficient Froude Number Reynolds Number Weber Number Morton Number Archimedes Number E¨tv¨s Number o o Grashof Number Inverse Viscosity Number Dimensionless film thickness cD =4gd  3u2 Fu Fr = gdRe = We = M= Ar = E¨ = o Gr = NF = + =F ud µF F u2 d gµF4 F 33F  21 1 µF2 g 2  2=F M1 2F gd2  d3 gF  µF21 3g 2 d 2 F µF  y=F Gr 1 2=  w F µFTwo Phase Flow Homogeneous 1D model for two phase flowSuitable average properties are determined and the mixture is treated as a single fluid in the equations of motion Nomenclature A cf D G j P P Q v v W x z  µ  w  Subscripts 1,2 12 m cross-sectional area of flow channel 2w friction factor ( .v2 ) hydraulic mean diameter ( 4A ) P mass flux ( W ) A superficial velocity ( Q ) A perimeter of flow chamber pressure volumetric flowrate specific volume ( 1 )  velocity in flow direction mass flowrate mass fraction of phase 2 (may be a function of z) length in flow direction volume fraction of phase 2 viscosity density wall shear stress angle of flow relative to horizontal phases phase 2 relative to phase 1 mixtureSome average properties for steady homogeneous flow m = .2 + (1 - ).1 j2 j1 .µ1 + .µ2 j j 1 x 1-x = + µm µ2 µ1 µm = Some useful equations mass flowrate mass flux volumetric flowrate superficial velocity W = W1 + W2 G = G1 + G2 Q = Q1 + Q2 Q1 Q2 + j = j1 + j 2 = A A or 1 x 1-x = + m 2 1 -Dukler -Mc Adamsactual velocityjF j2 v2 = 1-  G W v= = m Am W1 = Q1 .1 W2 = Q2 .2 v1 1 -  j1 = . j2 v2  G1 W1 1-x = = G2 W2 x vF =The equations of motion are:Continuity: W = m .v.A = const Momentum: W Energy: dP dv = -A - P.w - Am g.sin dz dzd v2 dqe dw - = W. (h + + gz) dz dz dz 2From the equation of motion dP P W dv = - .w - . - m g.sin dz A A dz 2cf m v2 dv = - - G. - m g.sin D dz 2 d W 2cf G - G. ( ) - m g.sin = - m D dz Am 2cf G2 W dA W d 1 = - - G( ( )- 2 ) - m g.sin m D A dz m A m dz 1 G2 1 dA 2cf G2 2 d -G ( )+ - m g.sin = - m D dz m m A dz =- =- =- 2cf G2 x 1 - x dx 1 1 d 1 d 1 x 1 - x 1 dA g.sin ( + )-G2 { ( - )+x ( )+(1-x) ( )}+G2( + ) - x D 2 1 dz 2 1 dz 2 dz 1 2 1 A dz ( 2 + 1-x ) 1 2cf G2 dx dv2 dv1 1 dA g.sin (v2 x+v1 (1-x))-G2 { (v2 -v1 )+x +(1-x) }+G2(v2 x+v1 (1-x)) - D dz dz dz A dz (v2 x + v1 (1 - x))rearranging in terms of22cf G2 dx dP dv2 dv1 1 dA g.sin (v2 x+v1 (1-x))-G2 { (v2 -v1 )+ .{x +(1-x) }}+G2 (v2 x+v1 (1-x)) - D dz dz dP dP A dz (v2 x + v1 (1 - x))dP dz g.sin (v2 x+v1 (1-x))1 fG - 2cD (v2 x + v1 (1 - x)) - G2 dx (v2 - v1 ) + G2 (v2 x + v1 (1 - x)) A dA - dP dz dz = 2 (x dv2 + (1 - x) dv1 ) dz 1 + G dP dP`

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