Theory of Multiphase Flows.

By: Yadav Gaurav NM.Tech Thermal Sciences

Project Guide: Dr. S. Jayraj

Pressure Drop Equation

• For single phase flows:

• The first term on the rhs is the frictional pressure gradient , second is the gravitational pressure gradient and last is the accleration pressure gradient

dz

Gudg

dA

ds

dz

dp )(sin

• Under normal circumstances fluid does not accelerate.

• Usually encountered in nozzles, diffusers because of change in area and also in compressible flows because there density changes with length.

• For heated tubes because liquid goes into vapour phase and that results in acceleration or deceleration to occur.

• For the two phase flows the pressure drop equation can be written in analogous manner as

• Here ρ is replaced by ρTP

)(sin))( 2211

21

1 uGuGdz

dg

dA

ds

dA

ds

dz

dpTP

TP

TP

1

• In 1 kg mixture : x kg gas and (1-x) kg liquid.

• In 1m3 of mixture: α m3of gas and (1- α) m3 of liquid.

lgTP xx )1(

lgTP )1(

Physical parameters delineating flow patterns

Surface Tension: • Keeps channel wall always wet with liquid

during G-L flows (unless they are heated.)• Tends to make small liquid droplets and small

gas bubbles spherical. Gravity:• In non vertical channel it tends to pull heavier

phase at the bottom.

Constant heat flux for vertical heated tube

• In this case the liquid film around the wall gets depleted due to vaporization and shear stress.

Difference b/w heated & unheated tube.• Amount of vapour and liquid changes along

the length. So flow distribution also changes.

• No churn flow in heated tubes.

• Droplet flows is present in heated tube which was not present in unheated tubes.

• There is presence of radial temperature profile in the tube.

• Continous change in phase fraction along the axial length.

• Departure from thermodynamic equilibrium.

Definition and Common Terminologies• Phase 1: Continous phase.• Phase 2: Discontinous or lighter phase.

Mass Flow Rate: W• W 1 ,W2

• WTP = W1 +W2 .

Volumetric Flow Rate: Q• Q=W/ ρ.• QTP=Q1+Q2

Density: ρ• ρ1, ρ2, ρTP .

Specific Volume:ν• ν1, ν2, νTP .

Viscosity:μ• μ1, μ2, μTP

C/S Area: A• A=A1+A2

Wetted Area/Wetted Perimeter: S• S1 = Area at which phase 1 is in contact with

wall.• S2 = Area at which phase 2 is in contact with

wall.• Si = Interfacial area.

Mass Flux: G=W/A= ρu.• G1=W1/A and G2=W2/A.

• GTP = G1+G2

Volume Flux: j = Q/A.• j1 = Q1/A and j2 = Q2/A

• jTP=j1+j2.

Velocity: u=Q/A• u1=Q1/A1 and u2=Q2/A2.• These are insitu velocity components.• It is difficult to measure A1 and A2 as they are

the local insitu areas and change along the axial length.

Superficial Velocity (Measurable Velocity):• u1s = Q1/A and u2s = Q2/A.• For 1D steady state flow , superficial velocity

and volumetric flux are same.

Volume Fraction: Inlet Volume Fraction:

• It is measurable quantity.

21

2

QQ

Q

Insitu Volume Fraction:

• It is not a measurable quantity.

Mass Fraction: x• It is quality x. A Measurable quantity.

21

2

21

2

AA

A

VV

V

21

2

WW

Wx

Relative Velocity: • u12 = u1-u2

• u21 = u2-u1

• u12 = -u21

Slip Ratio: k• k = u2/u1

• Also k = )).(1

).(1(

2

1

xx

Some Advanced Relations

• Q1 = j1A = u1A1 = u1A(1-α)

Therefore j1 = u1(1-α).

• Q1 = W1/ρ1 = G1A/ρ1 = {G(1-x)}.A/ρ1

Therefore j1 = {G(1-x)}./ρ1

• In same way j2 = u2α = Gx/ρ2

Drift Velocity: • It is the component velocity minus the

average velocity.• u1j = u1-j

• u2j = u2-j.

Drift Flux:• Volumetric flux of either component related

to a surface moving at volumetric avg velocity.• j21 = α(u2-j).

• j12 = (1- α)(u1-j)

Relation between Drift flux and component volumetric fluxes.

j21 = α(u2-j) = αu2 - αj

= αu2- αj1 -αj2

= j2- αj1 -αj2

= j2(1 -α) – αj1.

j12 = (1- α)(u1-j)

= (1- α)u1 - (1- α)j1 - (1- α)j2

=j1(α) – (1 - α)j2.

Therefore j12 = -j21

Relation between Drift Flux and relative velocity.

j21 = α(u2-j) = αu2 – αj

= αu2- αj1 -αj2

= αu2 – α(1- α)u1 – α2u2

= α(1- α)(u2-u1)

= α(1- α)u21.

Therefore Drift Flux is directly proportional to relative velocity for a particular system at particular condition i.e. α = c

Schematic layout of NCL