Basic Models

There are three main types of flow models in the analysis of two-phase flow pressure drops:

a) the `homogeneous’ flow model, b) the `separated’ flow model,

c) the `flow pattern’ or `drift flux’ models. In this more sophisticated approach, the two phases are considered to be arranged in one of three or four definite prescribed geometries. These geometries are based on the various

configurations or flow patterns found when a gas and a liquid flow together in a channel. The basic equations are solved within the framework of each of these idealized representations. In order to apply these models, it is

necessary to know when each should be used and to be able to predict the transition from one pattern to another.

The homogeneous model and the separated flow model are the two most widely

10²

4.1 The homogeneous flow model

The homogeneous model considers the two-phase flow as a homogenized mixture (pseudo-fluid) possessing mean fluid properties. It neglects any effect flow patterns may have on the two-phase flow and ignores interaction between the phases.

The basic assumptions of the homogeneous model are:

a) Equal vapor and liquid linear velocities U_g= U_f = U (slip ratio S = 1)

b) Thermal equilibrium exists between the two phases T_g= T_f=T_sat(p)

where T_sat(p) is the saturation temperature at the pressure p.

c) Frictional pressure drop can be calculated by substituting suitable weighted average values for properties such as velocity, density, and viscosity in the single-phase equations.

For the homogeneous flow model, the mixture density is given by

where ρ_his the mean density of the homogeneous fluid, and the void fraction is given by:

The ratio of the two-phase and single-phase pressure gradients is defined as the two-phase friction multiplier which, for the homogeneous flow model, can be expressed as:

where φ²_LO= two-phase friction multiplier for total flow assumed liquid.

f = single-phase friction factor f_tp= two-phase friction factor

In the homogeneous model, the underlying idea is to replace the two-phase fluid by an equivalent compressible single-phase fluid. The homogeneous model is applicable only if there is no rapid variation of flow parameters and its thermal non-equilibria have no great influence. Although the homogeneous model predicts the dependence of two-phase friction multiplier on pressure and quality reasonably well, it has two unsatisfactory features:

a. the friction multiplier is a function of pressure and quality only and is independent of mass flux, and

b. it generally underpredicts low quality data.

φ_LO² =f_tp

The effect of mass velocity on the two-phase friction multiplier has been widely reported. Experimental data from various sources show that there is a mass velocity effect with steam-water flow at high pressure. The higher the pressures and velocities, the more realistic the homogeneous model. The homogeneous model generally gives good agreement for mass fluxes greater than

2000 kg/m².s. Considering the various possible flow patterns which occur in two-phase flow, it is apparent that the theoretical concept of a homogeneous flow can be approached only in mist or finely dispersed bubbly flow.

4.2 The separated flow model

The separated flow model considers the two phases to be artificially segregated into two streams: one of liquid and one of vapor. Conservation equations are written separately for each phase and interaction between phases if taken into account by constitutive relationships. The basic equations for the separated flow model are not dependent on the particular flow configuration adopted. The basic assumptions of the separated flow model in the analysis of the two-phase pressure drop are:

a) The velocities of each phase are constant, but not necessarily equal, in any given cross-section, within the zone occupied by the phase.

b) Thermodynamic equilibrium exists between the two phases.

c) The use of empirical correlations to relate the two-phase friction multiplier φ² and the void fraction α to the dependent variables of the flow.

From a consideration of the various flow patterns it would be expected that this model would be most valid for the annular flow pattern.

The frictional pressure gradient can be expressed in terms of single-phase pressure gradient for the total flow considered as liquid.

Now, using the Blasius equation

The mean density of the two-phase fluid is:

Assuming that the gas phase and liquid phase pressure drops are equal, irrespective of the flow pattern, and the acceleration and elevation pressure drops are negligible then the two-phase multiplier for the separated flow model is given by:

In order to evaluate the frictional pressure gradient and the acceleration and elevation pressure drops it is necessary to develop expressions for the two-phase multiplier φ²_foand the void fraction α in terms of the independent flow

variables.

4.2.1 Martinelli-Nelson correlation

Martinelli-Nelson developed one of the earliest models for the two-phase pressure drop based on the basic assumptions of the separated flow model. The correlation was originally developed by Martinelli for round pipes with no heat addition. Martinelli-Nelson obtained values for the two-phase friction

multiplier and void friction as functions of pressure and mass quality based on the diabatic data of Davidson (20). A major shortcoming of the Martinelli-Nelson two-phase friction multiplier correlation is that it assumes the multiplier to be dependent on steam quality and pressure only and does not include the effect of mass velocity which has been observed experimentally. Figure 15 shows the two phase friction multiplier for various pressure and qualities.

Figure 15

Martinelli-Nelson two phase flow multiplier

1000

100

10

1

0 0.1 0.2 0.3 0.4

Quality

Two Phase Multiplier

0.5 0.6 0.7

16 MPa 8 MPa 4 MPa 2 MPa 1 MPa 0.5 MPa

4.2.2 Thom correlation

Based on high pressure steam, water pressure drop data obtained in horizontal and vertical tubes in diabatic as well as adiabatic conditions, Thom obtained a consistent set of values for the two-phase friction multiplier and void fraction in terms of pressure and quality.

This correlation predicts high pressure experimental data better than the Nelson. However, it has the same shortcoming as that of Martinelli-Nelson in that it does not include the effect of mass flow on the two-phase friction multiplier correctly.

And

4.3 The drift flux model

This model has been developed principally by Zuber (23) and Wallis (24) together with their co-workers.

The essential relationships of the drift flux model are presented below.

Relative velocity between the phases, U_gf,can be expressed as:

A drift flux, j_gf, is defined as:

The drift flux j_gf, physically represents the volumetric rate at which vapor is passing forwards (in up-flow) or backwards (in down-flow) through unit area of a plane normal to the channel axis already travelling with the flow at a velocity j.

To preserve continuity an equal and opposite drift flux of liquid (j_fg) must also pass across this same plane.

Rearranging,

The above relationship is true for one-dimensional flow or at any local point in the flow. It is often desirable to relax the restriction of one-dimensional flow.

j_g= αj + j_gf

Denoting the average properties of the flow by a bar, thus, u—, then, Define a parameter, C_o, such that,

It is also found convenient to define a weighted mean drift velocity,

It will be seen that if there is no local relative motion between the phases

for one-dimensional homogeneous flow, α = β. In other words the parameter C_o represents an empirical factor correcting the one-dimensional homogeneous theory to account for the fact that the concentration and velocity profiles across the channel can vary independently of one another.

The drift flux model can be used with or without reference to any particular flow regime.

The particular values these parameters take up will vary depending upon whether or not the chosen is restricted to a particular flow pattern.

Experimental data plottedas j_g / α vs j

are used toyield expressions for COand j_gf or alternatively ugj

u_g= 0 then α = βC_O or

α = β C_O + u_g

j ugj=j_gf

α then ug=j_g

α= COj + ugj

CO= αj α j j_g= αj + j_gf

The drift flux approach, therefore, satisfactorily accounts for the influence of mass velocity on the void fraction as seen in the separated flow model and equation and an empirical expression for C_omay be used to provide the required relationship between void fraction and the independent flow pattern.

The drift flux model is valuable only when the drift velocity is significant compared with the total volumetric flux say ( u–

gi> 0.05 j).