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Department of Chemical Engineering Fluids 4 ­ Two Phase Flow

Two 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 "One-Dimensional Two-Phase Flow" by G B Wallis, shows a plot of the typical flow regimes for vertical gas-liquid flows in a pipe.

jG m/s

10

de-wetting small ripples

Annular Churn

1

Impossible

Flooding

Slug

jF m/s Bubble

-1

Annular

0.1 -0.1

Bubble

0.01 0.1 1.0

jF m/s

-1

Annular

-10

Impossible

de-wetting small ripples

jG m/s

Map 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 Flow

1) At low jG we have Bubble Flow jG =

QG A

j

G

jF =

QF A

Bubble 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.

j

F

3) 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 thickness

j

G

j

F

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 >> film velocity. Droplets are entrained (as much as half of the liquid is entrained at high jG ). 5) Eventually we have Mist Flow

j

F

j j

F

G

­ fine droplets of liquid dispersed in the gas phase ­ some liquid still remains on the wall.

j

G

Countercurrent Vertical Flow

Only 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 Flow

Dispersed Bubble 3

j

F

1

Plug

Slug

m/s 0.2 0.1 Wave Stratified Annular Mist

0.01 0.1 1.0 4 10 j

G

100 m/s

Again there are a range of flow regimes dependant on jG and jf 1) Dispersed Bubble Flow

j

F

j

G

2) Plug Flow ­ elongated bubble

3) Slug Flow

4) Stratified Flow

gas

liquid

5) Wave Flow

gas

liquid

6) Annlar Mist

j

G

j

F

Analysis of two-phase flows

How 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 theory

2) Separated flow model

3) Drift flux model

Two Phase Flow Dimensionless Groups

NB = 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 F

u Fr =

gd

Re = We = M= Ar = E¨ = o Gr = NF = + =

F ud µF F u2 d

gµF4 F 3

3

F 2

1 1 µF2 g 2 2

=

F M

1 2

F gd2 d3 gF µF2

1 3

g 2 d 2 F µF y

=

F Gr

1 2

=

w F µF

Two Phase Flow Homogeneous 1D model for two phase flow

Suitable 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 mixture

Some 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 velocity

jF 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 dz

d v2 dqe dw - = W. (h + + gz) dz dz dz 2

From 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 of

2

2cf 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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