Void Fraction Model for Two-Phase Flow

To obtain the total pressure drop, accelerational and gravitational pressure drops must also be determined. As can be seen from

$-\frac{dp}{dz}= \frac{4{{\tau }_{w}}} {D}+\frac{\partial( {\dot{{m}''}}^{2} / \rho )}{\partial z}+\rho g\cos \theta$
$-\frac{dp}{dz}=\frac{4{{\tau }_{w}}}{D}+\dot{{{m}''}}\frac{d}{dz}\left[ \frac{{{x}^{2}}}{{{\rho }_{v}}\alpha }+\frac{{{(1-x)}^{2}}}{{{\rho }_{\ell }}(1-\alpha )} \right]+g\rho \cos \theta$

from Separated Flow Model for Two-Phase Flow, knowledge of the void fraction is required for determination of the accelerational and gravitational pressure drops for both the homogeneous and separated flow models. In the case of a horizontal circular tube, the gravitational pressure drop term becomes zero but the accelerational pressure drop terms are still present. Therefore, correlations for the void fraction in the two-phase flow will be discussed.

When the phase velocities differ for the liquid and vapor phases, the slip ratio defined in eq.

$S=\frac{{{w}_{v}}}{{{w}_{\ell }}}$

is frequently used in lieu of the void fraction. Substituting

${{G}_{\ell }}=\frac{{{{\dot{m}}}_{\ell }}}{A}={{\rho }_{\ell }}{{j}_{\ell }}={{\rho }_{\ell }}{{w}_{\ell }}(1-\alpha )$ and ${{G}_{v}}=\frac{{{{\dot{m}}}_{v}}}{A}={{\rho }_{v}}{{j}_{v}}={{\rho }_{v}}{{w}_{v}}\alpha$

into $S=\frac{{{w}_{v}}}{{{w}_{\ell }}}$, one obtains

$S=\frac{{{G}_{v}}{{\rho }_{\ell }}(1-\alpha )}{{{G}_{\ell }}{{\rho }_{v}}\alpha }\qquad\qquad(1)$

which can be simplified by using $x=\frac{{{G}_{v}}}{{\dot{{m}''}}}$, i.e.,

$S=\frac{{{\rho }_{\ell }}x(1-\alpha )}{{{\rho }_{v}}(1-x)\alpha }\qquad\qquad(2)$

The slip ratio can also be related to the volumetric flow rate by substituting

${{\left\langle {{w}_{\ell }} \right\rangle }^{\ell }}=\frac{{{Q}_{\ell }}}{{{A}_{\ell }}}$ and ${{\left\langle {{w}_{v}} \right\rangle }^{v}}=\frac{{{Q}_{v}}}{{{A}_{v}}}$

into eq. $S=\frac{{{w}_{v}}}{{{w}_{\ell }}}$, i.e.,

$S=\frac{{{Q}_{v}}{{A}_{\ell }}}{{{Q}_{\ell }}{{A}_{v}}}\qquad\qquad(3)$

Substituting

$\alpha =\frac{\Delta z\int_{{{A}_{v}}}{dA}}{\Delta z\int_{A}{dA}}=\frac{{{A}_{v}}}{{{A}_{v}}+{{A}_{\ell }}}$

into eq. (3), one obtains

$S=\frac{{{Q}_{v}}(1-\alpha )}{{{Q}_{\ell }}\alpha }\qquad\qquad(4)$

The relationships between the void fraction and slip ratio can be obtained by rearranging eqs. (2) and (4):

$\alpha =\frac{1}{1+\frac{1-x}{x}\frac{{{\rho }_{v}}}{{{\rho }_{\ell }}}S}\qquad\qquad(5)$
$\alpha =\frac{{{Q}_{v}}}{S{{Q}_{\ell }}+{{Q}_{v}}}\qquad\qquad(6)$

It can be observed from eqs. (2) and (4), (5), and (6) that the void fraction may be determined if the slip ratio is known, and vice versa. The void fraction in the homogeneous model obtained by

$\alpha =\frac{x}{x+(1-x){{\rho }_{v}}/{{\rho }_{\ell }}}$

from Homogeneous Flow Model for Two-Phase Flow is a special case of eq. (5) where the slip ratio S = 1. The homogeneous model is the better choice in a two-phase application for bubbly or mist flow, since the liquid-vapor interface is very small and the slip ratio is very close to unity. For other flow patterns where the velocity of the vapor phase is significantly higher than the velocity of the liquid phase, the homogeneous model tends to significantly overpredict the void fraction. Another case in which the homogeneous model will provide good results is near critical point (${{\rho }_{\ell }}\approx {{\rho }_{v}}$), in which case $\alpha \approx x$.

To predict the void fraction for cases where the velocities of the two phases differ significantly, correlations based on the separated flow model should be employed. The earliest void fraction correlation is based on the separated flow model proposed by Lockhart and Martinelli (1949), shown in Fig. 1 from Correlations Based on the Separated Flow Model. The relationship between the void fraction and the Martinelli parameter, X, is fitted by the following equation:

$\alpha =\frac{{{\phi }_{\ell }}_{,tt}-1}{{{\phi }_{\ell ,tt}}}\qquad\qquad(7)$

where ${{\phi }_{\ell }}_{,tt}$ is the frictional multiplier for turbulent flow in both vapor and liquid phases, obtained from

$\phi _{\ell }^{2}=1+\frac{C}{X}+\frac{1}{{{X}^{2}}}$

from Correlations Based on the Separated Flow Model with the appropriate value of C from Table 1 from Correlations Based on the Separated Flow Model, i.e.,

$\phi _{\ell ,tt}^{2}=1+\frac{20}{X}+\frac{1}{{{X}^{2}}}\qquad\qquad(8)$

Butterworth (1975) recommended the following simpler correlation to replace eq. (7):

$\alpha ={{[1+0.28{{X}^{0.71}}]}^{-1}}\qquad\qquad(9)$

The void fraction correlation proposed by Lockhart and Martinelli (1949) is the most widely-used correlation. However, it tends to overpredict the void fraction at high mass flow rate. A number of alternative correlations have been proposed to overcome this problem. The correlation proposed by Premoli et al. (1971) is worthwhile to introduce here because it covers a wide range of data. The correlation of Premoli et al. (1971) given in terms of the slip ratio is

$S=1+{{E}_{1}}{{\left[ \left( \frac{y}{1+y{{E}_{2}}} \right)-y{{E}_{2}} \right]}^{\frac{1}{2}}}\qquad\qquad(10)$

where

$y=\frac{\beta }{1-\beta }\qquad\qquad(11)$
${{E}_{1}}=1.578\operatorname{Re}_{\ell 0}^{-0.19}{{\left( \frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right)}^{0.22}}\qquad\qquad(12)$
${{E}_{2}}=0.0273\text{We}\operatorname{Re}_{\ell 0}^{-0.51}{{\left( \frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right)}^{-0.08}}\qquad\qquad(13)$

where β is the volumetric flow fraction of vapor flow defined by

$\beta =\frac{{{j}_{v}}}{{{j}_{\ell }}+{{j}_{v}}}=\frac{{{Q}_{v}}}{{{Q}_{\ell }}+{{Q}_{v}}}$

The Weber number in eqs. (12) and (13) is defined as

$\text{We}=\frac{{{{\dot{{m}''}}}^{2}}D}{{{\rho }_{\ell }}\sigma }\qquad\qquad(14)$

The void fraction can be obtained from eq. (5) after the slip ratio is obtained from eq. (10).

Butterworth (1975) compared different void fraction correlations for two-phase flow and suggested that the void fractions can be expressed in the following form

$\alpha =\frac{1}{1+c{{\left( \frac{x}{1-x} \right)}^{q}}{{\left( \frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{r}}{{\left( \frac{{{\mu }_{\ell }}}{{{\mu }_{v}}} \right)}^{s}}}\qquad\qquad(15)$
Table 1: Constants and exponents for different void fraction models.
 Models c q r s Homogeneous model 1 1 1 0 Zivi (1964) 1 1 0.67 0 Turner (1966) 1 0.72 0.40 0.08 Lockhart-Martinelli (1949) 0.28 0.64 0.36 0.07 Thome (1964) 1 1 0.89 0.18 Baroczy (1965) 1 0.74 0.65 0.13

where the values of the constant and exponents are given in Table 1. Chisholm (1973) presented a simple correlation in terms of slip-ratio

$S={{\left[ 1-x\left( 1-\frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right) \right]}^{0.5}}\qquad\qquad(16)$

Based on the assumption that both liquid and vapor are turbulent (both ${{\operatorname{Re}}_{\ell }}$ and Rev are greater than 2000), Awad and Muzychka (2005b) developed rational bounds for two-phase void fraction. The lower bound of the void fraction is

${{\alpha }_{lower}}={{\left\{ 1+{{\left[ {{\left( \frac{x}{1-x} \right)}^{0.875}}{{\left( \frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{0.5}}{{\left( \frac{{{\mu }_{\ell }}}{{{\mu }_{v}}} \right)}^{0.125}} \right]}^{16/19}} \right\}}^{-1}}\qquad\qquad(17)$

The upper bound of the void fraction is

${{\alpha }_{upper}}={{\left\{ 1+0.28{{\left[ {{\left( \frac{x}{1-x} \right)}^{0.875}}{{\left( \frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{0.5}}{{\left( \frac{{{\mu }_{\ell }}}{{{\mu }_{v}}} \right)}^{0.125}} \right]}^{0.71}} \right\}}^{-1}}\qquad\qquad(18)$

By averaging the lower and upper bounds, an empirical correlation for void fraction in two-phase flow can be obtained.

\begin{align} & {{\alpha }_{ave}}=\frac{1}{2}{{\left\{ 1+{{\left[ {{\left( \frac{x}{1-x} \right)}^{0.875}}{{\left( \frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{0.5}}{{\left( \frac{{{\mu }_{\ell }}}{{{\mu }_{v}}} \right)}^{0.125}} \right]}^{16/19}} \right\}}^{-1}} \\ & \text{ }+\frac{1}{2}{{\left\{ 1+0.28{{\left[ {{\left( \frac{x}{1-x} \right)}^{0.875}}{{\left( \frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{0.5}}{{\left( \frac{{{\mu }_{\ell }}}{{{\mu }_{v}}} \right)}^{0.125}} \right]}^{0.71}} \right\}}^{-1}} \\ \end{align}\qquad\qquad(19)

References

Awad, M.M., and Muzychka, Y.S., 2005b, “Bounds on Two-Phase Flow: Part II Void Fraction in Circular Tubes,” Proceedings of International Mechanical Engineering Congress and Exposition, Orlando, FL (DVD).

Baroczy, C.J., 1965, “Correlation of Liquid Fraction in Two-Phase Flow with Applications to Liquid Metals,” Chemical Engineering Progress Symposium Series, Vol. 61, pp. 179-191.

Butterworth, D., 1975, “A Comparison of Some Void-Fraction Relationships for Cocurrent Gas-Liquid Flow,” International Journal of Multiphase Flow, Vol. 1, pp. 845-80.

Chisholm, D., 1973, “Void Fraction During Two-Phase Flow,” Journal of Mechanical Engineering Science, Vol. 15, pp. 235-236.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Lockhart, R.W., and Martinelli, R.C., 1949, “Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes,” Chemical Engineering Progress Symposium Series, Vol. 45, pp. 39-48.

Premoli, A., Francesco, D., Prina, A., 1971, “A Dimensionless Correlation for Determining the Density of Two-Phase Mixtures,” Thermotecnica, Vol. 25, pp. 17-26.

Thome, J.R.S., 1964, “Prediction of Pressure Drop during Forced Circulation Boiling of Water,” International Journal of Heat and Mass Transfer, Vol. 7, pp. 709-724.

Turner, J.M., 1966, Annular Two-Phase Flow, Ph.D. Dissertation, Dartmouth College, Hanover, NH.

Zivi, S.M., 1964, “Estimation of Steady-State Void Fraction by Means of the Principle of Minimum Energy Production,” ASME Journal of Heat Transfer, Vol. 86, pp. 247-252.