Multiphase flow

Sediment transport in rivers is subject to multiphase flow, in which the suspended particles are treated as a disperse second phase which interacts with the continuous fluid phase.

An example of multiphase flow on a smaller scale would be within porous structures. Pore-structure modelling enables the use Darcy's law to calculate the volumetric flow rate through porous media such as groundwater flow through rock.[12] Further examples occur within the bodies of living organisms, such as blood flow (with plasma being the liquid phase and red blood cells constituting the solid phase.[13] Also flow within the intestinal tract of the human body, with solid food particles and water flowing simultaneously.[14]

The large majority of processing technology involves multiphase flow. A common example of multiphase flow in industry is a fluidized bed. This device combines a solid-liquid mixture and causes it to move like a fluid.[15] Further examples include water electrolysis,[16] bubbly flow in nuclear reactors, gas-particle flow in combustion reactors and fiber suspension flows within the pulp and paper industry.[17]

In oil and gas industries, multiphase flow often implies to simultaneous flow of oil, water and gas. The term is also applicable to the properties of a flow in some field where there is a chemical injection or various types of inhibitors.[18][19] In petroleum engineering, drilling fluid consists of a gas-solid phase. Furthermore, crude oil during flow through pipelines is a gas-oil-water three phase flow.[10]

Flow regimes in horizontal flow from top to bottom: Bubble flow, Plug flow, Slug flow, Wavy flow, Stratified flow, Annular flow and Mist flow

Flow patterns in pipes are governed by the diameter of the pipe, the physical properties of the fluids and their flow rates. As velocity and gas-liquid ratio is increased, "bubble flow" transitions into "mist flow". At high liquid-gas ratios, liquid forms the continuous phase and at low values it forms the disperse phase. In plug and slug flow, gas flows faster than the liquid and the liquid forms a 'slug' which becomes detached and velocity decreases until the next liquid slug catches up.[3]

Common vertical flow regimes - From left to right: Churn flow, Annular flow and Wispy annular flow

In Vertical flow axial symmetry exists and flow patterns are more stable.[2] However, in regards to slug flow oscillations in this regime can occur. Horizontal flow regimes can be applied here, however, we see a more even distribution of particles due to the buoyancy force acting in the direction of the pipe.

Churn flow occurs when slug flow breaks down, leading to an unstable regime in which there is an oscillatory motion of the liquid.

Wispy annular flow is characterised by the liquid 'wisps' that exist in the annular flow regime. Presumably due to the coalescence of the large concentration of contained droplets in the liquid film covering the pipe. This regime occurs at high mass fluxes.[9]

Hydraulic transport consists of flows in which solid particles are dispersed in a continuous liquid phase. They are often referred to as slurry flows. Applications include the transport of coals and ores to the flow of mud.[1]

Suspensions are classified into the following groups; fine suspensions in which the particles are uniformly distributed within the liquid and coarse suspensions where particles ted to travel predominantly in the bottom half of a horizontal pipe at a lower velocity than the liquid and a significantly lower velocity than the liquid in a vertical pipe.[3]

Gas–solid two-phase flow widely exists in chemical engineering,[29] power engineering, and metallurgical engineering. In order to reduce atmospheric pollution and pipe erosion, improve product quality, and process efficiency, the flow parameters measurement of two-phase flow by pneumatic conveying (using pressurised gas to induce flow) is becoming increasingly widespread.[30]

Three-phase flows are also of practical significance, and examples are as follows:

1. Gas-liquid-solid flows: this type of system occurs in two-phase fluidised bed and gas lift chemical reactors where a gas-liquid reaction is promoted by solid catalyst particles suspended in the mixture. Another example is in froth flotation as a method to separate minerals and carry out gas-liquid reactions in the presence of a catalyst[9]
2. Three-phase, gas-liquid-liquid flows: mixtures of vapors and two immiscible liquid phases are common in chemical engineering plants. Examples are gas-oil-water flows in oil recovery systems and immiscible condensate-vapor flows in steam/hydrocarbon condensing systems.[20] Further examples lie in the flow of oil, water and natural gas. These flow can occur in condensation or evaporation of liquid mixtures (e.g. the condensation or evaporation of steam or hydrocarbons)[9]
3. Solid-liquid-liquid flows: An example being sand mixing with oil and water in a pipeline[9]

Multiphase flows are not restricted to only three phases. An example of a four phase flow system would be that of direct-contact freeze crystallization in which, for example, butane liquid is injected into solution from which the crystals are to be formed, and freezing occurs as a result of the evaporation of the liquid butane. In this case, the four phases are, respectively, butane liquid, butane vapor, solute phase and crystalline (solid) phase.[20]

Due to the presence of multiple phases, there are considerable complications in describing and quantifying the nature of the flow compared with conditions of single phase flow. Velocity distribution is difficult to calculate due to the lack of knowledge of the velocities of each phase at a single point.

There are several ways to model multiphase flow, including the Euler-Langrange method, where the fluid phase is treated as a continuum by solving the Navier-Stokes equations. The dispersed phase is solved by tracking a large number of disperse particles, bubbles or droplets. The dispersed phase can exchange momentum, mass and energy with the fluid phase.[1]

Euler-Euler two phase flow is characterised by the volume-averaged mass conservation equation for each phase.[4] In this model, the disperse and continuous phase are treated as fluids. The concept of a volume fraction is introduced for each phase, discussed in the parameter section below.

The most simple method to categorize continuous multiphase flows is to consider treat each phase independently. This concept is known as the homogeneous flow model first proposed by Soviet scientists in the 1960s. Assumptions in this model are:

• The gas phase velocity is equal to the liquid phase velocity
• Two-phase medium is in thermodynamic equilibrium[11]

For multiphase flow in pipes, the mass flow rate for each phase can be determined using the equation:

${\displaystyle G={\dot {m}}=\lim \limits _{\Delta t\rightarrow 0}{\frac {\Delta m}{\Delta t}}={\frac {{\rm {d}}m}{{\rm {d}}t}}}$

Where ${\displaystyle \ G}$ = mass flow rate of a single phase, Δ = change in quantity, m = Mass of that phase t = time and the dot above m being a time derivative.[31]

The volumetric flow rate can be described using the below equation:

${\displaystyle Q={\dot {V}}=\lim \limits _{\Delta t\rightarrow 0}{\frac {\Delta V}{\Delta t}}={\frac {\mathrm {d} V}{\mathrm {d} t}}}$

Where Q = volumetric flow rate of a single phase, V = Volume.[1]

The variables stated above can be input into the below parameters that are important in the description of multiphase flow. In wellbore multiphase flow the mass flow rate, volumetric fraction and velocity of each phase are important parameters.[11]

Key parameters that describe multiphase flow in a conduit.[11]

Parameter	Equation	Description
Mass flow rate	${\displaystyle G=G_{g}+G_{l}+G_{s}}$	Mass flow rate being the mass of fluid which passes through the cross-section per unit of time. Where G = mass flow rate, g = gas, l = liquid and s = solid.
Volumetric flow rate	${\displaystyle Q=Q_{g}+Q_{l}+Q_{s}}$	The Volumetric flow rate, defined as the volume of fluid passing through a cross sectional area per unit of time:
Mass fraction	${\displaystyle x_{i}=\lim \limits _{\delta V\rightarrow V^{o}}{\frac {\Delta G_{i}}{\Delta G}}={\frac {G_{i}}{G_{g}+G_{l}+G_{s}}}}$	Where Gi is the mass flow rate of either the liquid, solid or gas phase. Defined as the ratio of the mass of one phase to the total mass of the mixture passing through the cross-section per unit of time.
Volume fraction	${\displaystyle \beta _{i}=\lim \limits _{\delta V\rightarrow V^{o}}{\frac {\Delta Q_{i}}{\Delta Q}}={\frac {Q_{i}}{Q_{g}+Q_{l}+Q_{s}}}}$	Where Qi is the volumetric flow rate of either the liquid, solid or gas phase. Q is the total volumetric flow rate. The volume fraction is defined as the ratio of the volume of one phase divided by the total volume of the mixture passing through the cross-section per unit of time.[1]
Superficial velocity	${\displaystyle s_{g}={\frac {Q_{g}}{A}}}$ ${\displaystyle s_{l}={\frac {Q_{l}}{A}}}$ ${\displaystyle s_{s}={\frac {Q_{s}}{A}}}$	Where ${\displaystyle s_{g}=}$ superficial velocity of gas phase (m/s), ${\displaystyle s_{l}=}$velocity of liquid phase and ${\displaystyle s_{s}=}$velocity of solid phase. Superficial velocity is a hypothetical velocity wherein the assumption is that one phase occupies the entire cross sectional area.
Actual velocity	${\displaystyle v_{g}={\frac {Q_{g}}{A_{g}}}}$ ${\displaystyle v_{l}={\frac {Q_{l}}{A_{l}}}}$ ${\displaystyle v_{s}={\frac {Q_{s}}{A_{s}}}}$	Where ${\displaystyle v_{g}=}$ actual velocity of gas phase (m/s), ${\displaystyle v_{l}=}$velocity of liquid phase and ${\displaystyle v_{s}=}$velocity of solid phase.

A flow through a conduit of constant cross-sectional area is considered to be under steady-state conditions when its velocity and pressure may vary from point to point but do not change with time. If these conditions are variable with time then the flow is known as transient.[11] The gas phase most commonly flows at a higher velocity than the liquid phase, this is due to the lower density and viscosity.[3]

The volumetric flow rate and fluid motion, in general, is driven by different forces acting on fluid elements. There are five forces that affect flow rate, each of these forces can be categorised in three different types; line, surface and volume.

Consider a line element of length L on Volume forces act on an element proportional to the volume (${\displaystyle V\propto L^{3}}$). Surface forces act on elements proportional to the size of the area (${\displaystyle A\propto L^{2}}$) and line forces act on one dimensional curve elements (${\displaystyle \zeta \propto L}$):

Classifying the forces involved in multiphase flow

Force	Type	Magnitude of force	Magnitude of force per unit volume
Pressure	Surface	${\displaystyle F_{p}\propto \mathrm {A} \Delta P}$	${\displaystyle f_{P}\propto \Delta PL^{-1}}$
Inertia	Volume	${\displaystyle F_{I}\propto V\Delta \rho U^{2}L^{-1}}$	${\displaystyle f_{I}\propto \rho U^{2}L^{-1}}$
Viscous	Surface	${\displaystyle F_{V}\propto \mathrm {A} \mu UL^{-1}}$	${\displaystyle f_{V}\propto \mu UL^{-2}}$
Buoyancy	Volume	${\displaystyle F_{B}\propto \ Vg\Delta \rho }$	${\displaystyle f_{B}\propto g\Delta \rho }$
Surface	Line	${\displaystyle F_{S}\propto \ L\sigma }$	${\displaystyle f_{S}\propto \sigma L^{-2}}$

Where P = pressure, ρ = mass density, Δ = change in quantity, σ = surface tension, μ = Dynamic viscosity, A = area g = acceleration due to gravity, L = linear dimension, V = volume, U = velocity of continuous phase.[32]

The pressure force acts on an area or surface elements and accelerates the fluid in the downwards direction of the pressure gradient. The pressure difference between the beginning and the end of the pressure gradient is known as the pressure drop. The Darcy-Weisbach equation can be utilised to calculate pressure drop in a channel.

The viscous force acts on a surface or area element and tends to make the flow uniform by diminishing velocity differences between phases, effectively opposes flow and lessens flow rate. This is evident in comparisons between high viscosity oil mixtures in comparison with low viscosity mixtures, where the higher viscosity oil moves slower.[33]

The inertia force is a volume force, which retains the direction and the magnitude of the motion. It is equivalent to the magnitude of the mass of the element multiplied by its acceleration. Acceleration is defined in this case as ${\displaystyle U^{2}L^{-1}}$, due to the linear dimension L being proportional to time. Higher inertia forces lead to turbulence, whereas lower inertia results in laminar flow.

The buoyancy force represents the net action of gravity whilst the density is non-uniform. The surface-tension force acts on a line or curve element and minimizes the surface area of the interface - this force is specific to gas-liquid or liquid-liquid flows.[32]

A vortex street around a cylinder which can occur in multiphase flow. This occurs around a Reynolds number between 40 and 1000 independent of cylinder size, fluid speed and fluid.[2]

From the forces shown in the table above, five independent dimensionless quantities can be derived, these relations provide insight into how the multiphase flow will behave:

The Reynolds number. This number predicts whether flow in each phase is either turbulent or laminar.

${\displaystyle \mathrm {Re} ={\frac {F_{I}}{F_{V}}}={\frac {f_{I}}{f_{V}}}={\frac {\rho \ LU}{\mu }}}$

At low Reynolds numbers, flow tends towards laminar flow, whereas at high numbers turbulence results from differences in fluid speed.

In general, laminar flow occurs when Re < 2300 and turbulent flow occurs when Re >4000. In the interval, both laminar and turbulent flows are possible and these are called transition flows. This number is dependent on geometry of the flow.[34]

For a mixture of oil and water flowing at high velocity it is most common to form a dispersed bubble type flow. Turbulent flow consists of eddies of different size range. Eddies that have larger size than droplets, transport these droplets through the flow field. Eddies, which are smaller or equal to the size of the droplets, cause droplet deformation and break-up. It can be viewed as eddies collide with droplets and break them if they have sufficient energy to overcome the droplets internal forces.

At the same time, turbulent flow induces droplet-droplet interaction, which is important for the coalescence mechanism. When two droplets collide, this may lead to coalescence, resulting in a bigger droplet size.

The Euler number describes the relationship between the pressure and inertial forces.

${\displaystyle \mathrm {Eu} ={\frac {F_{P}}{F_{I}}}={\frac {f_{P}}{f_{I}}}={\frac {\Delta \ p}{\ \rho U^{2}}}}$

It is used to characterise energy losses in the flow. A completely friction-less flow is represented by an Euler number of 1. This number is important when the pressure force is dominant. Examples include, flow through pipes, flow over submerged bodies and flow of water through orifices.

The Froude number is the ratio of inertia against gravity.

${\displaystyle \mathrm {Fr} ={\frac {F_{I}}{F_{G}}}={\frac {f_{I}}{f_{G}}}={\frac {U^{2}}{gL}}}$

When Fr < 1, small surface waves move upstream, Fr > 1 they will be carried downstream and when Fr = 0 the velocity is equal to the surface waves. This number is relevant when gravitational force is predominant in the fluid motion. For example, open channel flow, wave motion in the ocean, forces on bridge piers and offshore structures.

The Eötvös number defines the ratio of buoyancy compared with surface tension forces.

${\displaystyle \mathrm {Eo} ={\frac {F_{B}}{F_{S}}}={\frac {f_{B}}{f_{S}}}={\frac {\Delta \rho gL^{2}}{\sigma }}}$

A high value for this number indicates that the system is relatively unaffected by surface tension effects. A low value indicates that surface tension dominates.

The Weber number determines the relationship between the inertial force and surface tension.

${\displaystyle \mathrm {We} ={\frac {F_{I}}{F_{S}}}={\frac {f_{I}}{f_{S}}}={\frac {\rho LU^{2}}{\sigma }}}$

It also determines the droplet size of the disperse phase. This number is used extensively in flow regime maps. The influence of pipe diameter is well understood through the Weber number.

Three different regimes in assuming that gravity is negligible, or within microgravity can be identified:

1. Surface tension dominated regime with bubble and slug flow. (We<1)
2. An inertia dominated regime with annular flow. (We>20)
3. Transitional regime with a frothy slug-annular flow.

The transition from frothy slug-annular flow to fully developed annular flow occurs at We = 20.

The Capillary number can be defined using the Weber number and the Reynolds number. It is the relative importance of viscous forces relative to surface forces.

${\displaystyle \mathrm {Ca} ={\frac {F_{V}}{F_{S}}}={\frac {f_{V}}{f_{S}}}={\frac {\mu U}{\sigma }}={\frac {We}{Re}}}$

In microchannel flows, the capillary number plays a critical role as both surface tension and viscous forces are important.

In enhanced oil recovery operations, capillary number is an important number to consider. Whilst capillary number is higher, viscous forces dominate and the effect of interface tension between fluids in rock pores are reduced thereby augmenting recovery. In typical reservoir conditions, capillary number varies from 10⁻⁸ to 10⁻².[35]