Two Phase Flows
2. Two-Phase Flow Theory and Definitions
A full notation is given at the end of the lesson. This section introduces the primary variables used throughout this lesson and derives some simple relationships between them for one-dimensional flow.
To distinguish between gas and liquid, the subscripts “g” for gas and “f” for liquid (fluid) will be used. Consider a channel in which two phases are flowing cocurrently. The flow is one-dimensional: there are no changes in system properties in directions normal to the direction of flow. The flow area of the channel is denoted by A and the flow areas of the gas and liquid phases by Ag and Afrespectively. The mass rate of flow will be represented by the symbol W and will be the sum of the individual phase flow rates Wfand Wg. The velocity of the individual phases is denoted by the symbols Ugand Uf. The volumetric rate of flow is represented by the symbol Q and will be the sum of the individual volumetric flow rates Qfand Qg.
Two-phase flow takes several forms, shown in Figures 1 to 5.
Bubble flow is the case in which individual dispersed bubbles move independently up the channel (Figure 1).
Figure 1 Bubble flow
Plug, or slug flow is the case where patches of coalesced vapor fill most of the channel cross section as they move from inlet to outlet (Figure 2). Slug flow has been reported as being both a stable and an unstable transition flow between bubble flow and the next type, annular flow.
Figure 2 Slug flow
In annular flow the vapor forms a continuous phase, carrying only dispersed liquid droplets, and travels up the channel core, leaving an annulus of
superheated liquid adjacent to the walls as shown in Figure 3.
Figure 3 Annular flow
The fourth type shown in Figure 4 is called fog, dispersed, or homogeneous flow.
This is the opposite of the first type, bubble flow, in that in the latter case the vapor fills the entire channel and the liquid is dispersed throughout the vapor in the form of individual droplets.
Liquid film
Vapour Liquid
drops Liquid Vapour Liquid Vapour
bubbles
Figure 4 Fog flow
The fifth type, stratified flow, occurs at low flows in horizontal pipes when the steam and the water can separate due to gravity and buoyancy forces. That type of flow is of particular importance in Candu reactors because of their horizontal fuel channels and the potential for flow stratification when the primary system in not in nominal conditions.
Figure 5 Stratified flow
The paragraphs below introduce terms and definitions that characterize two phase flow such as quality, void, slip,...
2.1 Mass quality, x
The quality x of a vapor liquid mixture in a nonflow system, or where no relative motion between the vapor and liquid phases exists, is defined as:
x = mass of vapour in mixture total mass of mixture
It is often convenient to use the fraction of the total mass flow which is composed of vapor or liquid.
2.2 Void fraction, a The void fraction a is defined as:
The term void here is somewhat misleading, since there is actually no void but vapor.
α =volume of vapour in mixture total volume of mixture x= Wg
Wg+ Wf
Liquid Vapour
Liquid drops Vapour
The ratio of the sectional area occupied by the gas phase to the total cross-sectional area is defined as the void fraction:
and the non voided fraction as:
2.3 Relationship between quality and void
The relationship between x and α in a nonflow system can be obtained by assuming a certain volume containing 1 kg of mixture in thermal equilibrium as shown in Figure 6. That volume will be equal to υ = (1−x)υf+ xυg
where υ is the specific volume of the mixture.
Figure 6
Two phase mixture
In an equilibrium mixture, the two phases are saturated liquid and saturated vapor and the volume of vapor present is equal to its mass, x, times its specific volume υg and α is thus given by:
This equation can also be written in the form
where the specific volumes are all taken at the system pressure from appropriate thermodynamic property tables. Figure 7 shows calculated values for α versus x for light water at various pressures.
α = 1
1 + 1 – xx νf
νg
α = x νg
(1 – x) νf+ x νg
Liquid Vapour
bubbles 1 – α =Af
A α =Ag
A
Figure 7
Void versus quality for different pressures
Examination of the curves reveals the following:
1. For constant x, α decreases with pressure.
2. For any pressure, dα/dx decreases with x.
3. At low values of x, dα/dx increases as the pressure decreases and becomes very large at low pressure.
At atmospheric pressure, it can be noted that a small quality (about 2 %) generates almost 100 % void.
2.4 Mass Flux, G
The rate of mass flow divided by the flow area is given the name mass flux or mass velocity.
The gas velocity can be written as:
G = WA Wg= G A x and
Wf= G A 1 – x .1 MPa
1 MPa
10 MPa 15 MPa 100
90
80
70
60
50
40
30
20
10
00 5 10 15 20 25
Quality (%)
Void (%)
and the liquid velocity as:
where ρgand ρfare the densities of the vapor and the liquid phases.
2.5 Volumetric quality, ß
Sometimes, it is necessary to use the fraction of the total volumetric flow which is composed of vapor or liquid.
2.6 Volumetric flux, j
The volumetric flux or superficial velocity j is defined as the rate of volumetric flow divided by the flow area.
The liquid and gas volumetric fluxes are then defined as:
2.7 Slip ratio, S
In the above calculations it was assumed that no relative motion existed between the two phases. However, if a two-phase mixture is moving, the vapor, because of its buoyancy and its different resistance characteristics, has a tendency to slip past the liquid and move at a higher velocity than that of the liquid. This is particularly true in vertical sections. The homogeneous solution in which each phase is assumed to move at the same speed, however, has been found
satisfactory in most cases. In the homogeneous system, a slip ratio S, equal to 1 in nonflow or homogeneous flow and greater than 1 in nonhomogeneous two-phase systems is used. It is defined as the ratio of the average velocity of the vapor Ugto that of the liquid Uf. Thus the definition of the slip ratio:
The slip ratio modifies the relationship between void fraction and quality developed in the previous section. This will now be shown with the help of Figure 8 which shows a two-phase mixture flowing horizontally in a channel. A
S =Ug
certain section, between the dotted lines, small enough so that x and α remain unchanged, is considered.
Figure 8
Two phase mixture
In a flow system, the quality at any one cross section is defined by
Thus if the total mass flow of the mixture is W, the vapor-flow rate is xW and the liquid-flow rate is (1-x)W, where x is the quality at the particular section.
Applying the continuity equation, the velocities of vapor and liquid are given by:
where Agand Af are the cross-sectional areas of the two phases, perpendicular to flow direction, if the two phases are imagined to be completely separated from each other such as in Figure 9.
Figure 9
Separated two phase mixture
Fuel Vapour Ag At
Liquid
Ug=νgW Ag x and
Uf=νf 1 – x W Af
x= Mass flow rate of vapour Mass flow rate of mixture
Fuel elements
Liquid / Vapour Mix
Combining the above equations gives
The void fraction in the section considered is the ratio of the vapor-phase volume to the total volume within the section. In the small section of channel considered, this is the same as the ratio of cross-sectional area of vapor Agto the total cross-sectional area of the channel. Thus
The slip factor S can then be written as :
This equation can be rearranged to give a relationship between α and x, including the effect of slip, as:
By re-arranging the above expression, the following relationship for void fraction can be produced.
By re-arranging the above expression, the following relationship for volumetric quality can be produced.
Note that the relationship between α and x for no slip (S = 1) is a special case of the general relationship.
The effect of slip is to decrease the value of α corresponding to a certain value of x below that which exists for no slip. This can be seen in the slip equation.
β = 1 + 1 – x
At constant pressure and quality, the factor(1 - α)/α is directly proportional to S.
Thus, α decreases with S. A high S is thus an advantage from both the heat-transfer and moderating-effect standpoints. Figure 10 shows α versus x for light water at 100 bars and several slip ratios.
Figure 10
Void versus quality for different slip ratios
S has been experimentally found to decrease with both the system pressure and the volumetric flow rate and to increase with power density. It has also been found to increase with the quality at high pressures but to decrease with it at very low pressures.
As a function of the channel length, S has been found to increase rapidly at the beginning and then more slowly as the channel exit is approached. At the exit itself, turbulence seems to cause a sudden jump in the value of S.
Experimental data or theoretical correlations for S covering all possible operating and design variables do not exist. In boiling-reactor studies, values for S may be estimated from data that closely approach those of interest. In this a certain amount of individual judgment is necessary. Otherwise, experimental values of S under similar conditions of a particular design must be obtained. This
procedure is usually expensive and time-consuming but may be necessary in some cases. The importance of obtaining accurate values of S may best be emphasized by the following: One step in the procedure of core channel design
S=1
is to set a maximum value of α at the channel exit.
This is usually determined from nuclear (moderation) considerations. A corresponding value of x, at the selected S, is then found from the above equations. The latter determines the heat generated in the channel.
In design, the usual procedure is to assume a constant value of S along the length of the channel. This, of course, is a simplification, which may introduce further error into the results. However, S is seen to be fairly constant over most of the channel length, indicating this assumption to be a good one.
2.8 Two-Phase density, ρtp
The magnitude of the two-phase density, ρtp, depends on the specific void fraction relationship used for the calculation.