Annular flow is a flow regime of two-phase gas-liquid flow (see gas-liquid flow). It is characterized by the presence of a liquid film flowing on the channel wall (in a round channel this film is annulus-shaped which gives the name to this type of flow) and with the gas flowing in the gas core. The flow core can contain entrained liquid droplets. In this case, the region is often referred to as *annular-dispersed flow*, where the entrained fraction may vary from zero (a pure annular flow) to a value close to unity (a dispersed flow). Often both types of flow, pure annular and annular-dispersed, are known under the general term of annular flow.

In vertical and inclined channels, the near-wall liquid film and the gas core may be both concurrent and countercurrent. The true volumetric gas concentration (void fraction) of the annular flow ε_{G}, determined as a fraction of the cross section occupied by the gas phase, is high and, as a rule, exceeds 75-80%. In horizontal channels the film thickness is non-uniform around the channel perimeter due to gravity. Both adiabatic annular flows and diabatic annular flows occur in industrial applications: e.g., in steam generators, evaporators, condensers and boiling water reactors. In the latter case, a phase transition (evaporation, condensation) is observed at the liquid film-gas core interface. The phase change may also occur in the liquid film when vapor bubbles appear on the heated channel wall (nucleate boiling).

In the case of high wall superheats a regime of film boiling sets in when a vapor film flows along the wall and a liquid core flows in the middle of the channel. In this case, we are concerned with an *inverse annular flow*.

A general diagram of annular flow is presented in Figure 1 in which , , and are the flow rates of the gas (vapor) phase, the liquid in the wall film, and liquid entrained in the core respectively. The velocity profile in a gas core, as in the case in the flow of single-phase fluid, obeys the logarithmic law. However, as grows, it becomes more peaked which is related to both the increasing number of droplets in the core and a rougher film-core interface.

The thickness δ of the liquid film is determined by the ratio of
to
(the appropriate Reynolds numbers Re_{L} and Re_{G}) and by the shear stress τ_{i} on the interface (related to the pressure gradient in a two-phase flow dp_{f}/dx). The relation between δ,
and dp_{f}/dx is known as a *triangular relationship*. Given any two of the parameters, we can determine the unknown one making use of a model of shear stress distribution in the film (for fairly thin films the assumption τ = const is justified). Various forms of the triangular relationship are discussed by Hewitt and Hall-Taylor (1970) and by Hewitt (1982).

In order to determine ε_{G} in a pipe of diameter D for low ratios of working pressure to critical pressure the empirical *correlation of Armand* may be used

where

is the mass flux of the liquid, A the channel cross section, ρ_{L} and ρ_{G} are the densities, respectively, of the liquid and the gaseous phase, and η_{L} the dynamic viscosity of the liquid phase. The mean thickness δ of the film is related to ε_{G} by an obvious relation

The above formulas pertain to the case of concurrent phase motion. If we deal with a liquid film flowing down the channel wall and with the gas counterflow, then the moment comes, as the gas velocity increase, when on the film surface there are generated large waves which are carried unward by the gas stream, i.e., liquid transfer is observed upward of the place of its injection. This is known as flooding. As the gas velocity increases, the fluid flow reverses and the annular flow becomes concurrent, i.e., upward.

The film surface may be covered with an intricate *wave* system. The waves arise on a fairly thick film and make interface rough. Two types of waves are distinguished, viz. small-scale waves or ripples and large-scale or *disturbance waves* whose amplitude is five- or sixfold higher than the mean thickness of the film. The spacing between the disturbance wave crests is a function of a film thickness alone and does not depend on flow velocity. Experimental findings show that the disturbance waves are extremely nonuniform both axially and circumferentially. It is the disturbance waves, from the crests of which liquid breaks away, which are responsible for droplet entrainment to the core. *Droplet entrainment* may also occur as a result of bubbles bursbury from the film, giving rise a range of drop sizes.

The range of droplet sizes in the flow core is fairly broad. Their maximum size is determined by the critical value of the Weber number We = ρu^{2}_{r}D_{d}/ζ2σ, where u_{r} is the relative droplet velocity in the carrier vapor flow and D_{d} the droplet diameter. During acceleration of a droplet broken away from the surface it may be broken up. The final size distribution of the droplets depends on their initial size and the path traveled along the channel.

The droplets in the flow core deposits on the film surface. The deposition rate per unit peripheral as depends on a combined action of buoyancy, inertial, and Magnus forces. The behavior of the finest droplets depends on turbulent diffusion.

For estimating Hewitt and Hall-Taylor (1970) suggested the formula

where

is the mass concentration of droplets in the flow core. The deposition coefficient k_{D} has the dimensions of a velocity and is a function of fluid physical properties and flow rates. It also appears to depend on concentration of droplets at high concentrations. Correlations for k_{D} and for the rate of liquid entrainment in E are discussed by Hewitt and Govan (1990a and 1990b).

The relationship between
and quality, x, is illustrated in Figure 2. Straight line 1 shows the total mass of liquid which decreases linearly with increasing vapor quality x. Curve 2 describes entrainment in a long, adiabatic channel in which deposition and entrainment reach equality (
). Thus, at a quality denoted by C, the entrained droplet flow (M_{LE}) is represented by CB and the liquid film flow (M_{LF}) is represented by BA. Curve 3 describes entrainment in a channel with heating. In this case, the hydrodynamically equilibrium state is not attained. Here, the wall film is additionally depleted as a result of evaporation. With an allowance for the factor indicated above an overall mass balance is described by

where h_{LG} is the latent heat of vaporization and
the heat flux on the wall. It should be pointed out that, due to the presence of radial vapor flow from the film surface
may be somewhat lower than under adiabatic conditions. Conversely, because vapor bubbles burst from the liquid film
may increase compared to its value for adiabatic flow.

**Figure 2. Entrained liquid flow rate in adiabatic equilibrium (Curve 3) and diabatic (heated) flow (Curve 1).**

A complete analytical description of annular flow is beyond current capabilities. Most often, annular flow is calculated using one-dimensional approximation. In this case, the continuity equation becomes:

The momentum equation is:

and the energy equation is:

In the above equations u_{G}, u_{LE}, and u_{LF} are the mean velocities of gas phase, droplets, and liquid film, respectively; ε_{G}, ε_{LE}, and ε_{LF} are the fractions of the channel cross section occupied by an appropriate phase ( ε_{G} + ε_{LE} + ε_{LF} = 1 ). h_{0} = h + (u^{2}/2) is the stagnation enthalpy, z the axial distance, and θ the angle of inclination of the channel from the horizontal. dp/dz is the total pressure gradient and dp_{F}/dz is the frictional pressure gradient. These parameters are related to vapor quality of the flow

and the fraction of liquid phase entrained as droplets is given by

Other obvious relations are
, ρ_{L}u_{LE}ε_{LE} = m(1-x)ψ , ρ_{L}u_{LF}ε_{LF} = m(1 - x)(1 - ψ) where in is the lot. It should be borne in mind that the velocity of droplets u_{LE} differs slightly from u_{G} and substantially exceeds u_{LF}.

Equations (6) and (7) take into account mass and energy transfer between phases, external input of heat, change of phase fraction, and change of physical properties with pressure. The equations need to be supplemented by the relevant relations for physical properties and flow parameters.

It is obvious that in order to use Eqs. (5)-(7) we must know the values ε_{G}, ε_{LF}, ψ_{L} and dp_{F}/dz. Their authentic description can be obtained mainly from experimental data. In view of extremely diverse experimental conditions (geometrical, regime, kind of fluid), universal relations are not available to date while those available should be used with caution. The value ε_{G} can be estimated using Armand’s correlation (1). Alternatively, the liquid film thickness (and hence ε_{G}) can be determined from the liquid film flow rate and from the pressure gradient (the “triangular relationship - see Hewitt and Hall-Taylor, 1970 and Hewitt, 1982). Values of ψ may be estimated using the relationships of Hewitt and Govan (1990a, 1990b). The frictional pressure gradient dp_{F}/dz can be calculated from a relationship suggested by Wallis (1969) which takes account of the additional interfacial shear exerted by the gas in the liquid film as a result of interfacial waves:

where f_{a} is the friction factor for gas flow alone in the tube and in the gas mass flux (kg/m^{2} s calculated to the fuel correction. Heat transfer in annular flows may occur both with and without phase change.

Annular flow heat transfer with phase change (evaporation or condensation), is encountered in a wide range of industrial plant. Heat transfer in evaporation in the annular flow regime is noteworthy for high heat transfer coefficients. The mechanism of heat transfer may be nucleate boiling dominated, forced convection boiling dominated (i.e., without nucleation) or be a combination.

At given combinations of , p, x, and the wall liquid film is exhausted, the wall is dried out, and “dryout” or “burnout” sets in (see Burnout-forced Convection). This is accompanied with a rapid rise in wall temperature if the wall heat flux is fixed.

#### REFERENCES

Hewitt, G. F. and Hall-Taylor, N. S. (1972) Annular Two-Phase Flow. Pergamon Press, Oxford.

Hewitt, G. F. (1982) Chapter 2 of Handbook of Multiphase System. McGraw-Hill, New York.

Wallis, G. B. (1969) One-Dimensional Two-Phase Flow. McGraw-Hill Book Company, New York.

Hewitt, G. F. and Govan, A. H. (1990a) Phenomenological modelling of non-equilibrium flows with phase change. *Int. J. Heat Mass Transfer*, 133: 229-249. DOI: 10.1016/0017-9310(90)90094-B

Hewitt, G. F. and Govan, A. H. (1990b) Phenomena and prediction in annular two-phase flow. Invited Lecture, Symposium on Advances in Gas-Liquid Flow, ASME Winter Annular Meeting, Dallas, November 1990 (Published ASME Vol. 99/HTD Vol. 157, pp. 41-46).