Flooding or entrainment limit

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The flooding limit occurs due to the instability of the liquid film generated by a high value of interfacial shear, which is a result of the large vapor velocities. For example, in a vertical closed two-phase thermosyphon, the condensate liquid film flows down the walls and evaporates at the bottom. When a large velocity opposite to the liquid flow occurs due to evaporation, the flow of the condensate liquid film can stop. The vapor shear hold-up prevents the condensate from returning to the evaporator and leads to a flooding condition in the condenser section. This causes a partial dryout of the evaporator, which results in wall temperature excursions or in limiting the operation of the system. Flooding can also happen in cocurrent two-phase open systems.

There are two major fundamental semi-empirical correlations for the prediction of flooding limit of open two-phase systems. The first is the Wallis correlation (1969), which is characterized by a balance between the inertia and hydrostatic forces. The second is the Kutateladze two-phase flow stability criterion, in which the inertia, buoyancy, and surface tension forces are balanced (Kutateladze, 1972). The Wallis empirical correlation is based on results from open channel water-gas experiments. Two coefficients in the Wallis correlation must be determined by experiment because they are dependent upon the design of the pipe. The shortcoming of the Wallis correlation is that the effect of surface tension is not taken into account. Surface tension is of great importance to the hydrodynamic and heat transfer characteristics of gas-liquid systems. Physically, increasing the surface tension means that a higher pressure difference can be sustained across a film surface without forming waves. In the Kutateladze correlation, the effect of the diameter of the pipe is not included. For small tubes, the diameter of the vapor passage plays an important role in flooding characteristics. It was shown by Wallis and Makkenchery (1974) that the Kutateladze criterion produces a good correlation of the results for pipes with large diameters, but for small pipes the effect of the diameter should be considered. Various efforts have been made by investigators to extend the existing semi-empirical correlation from two-phase open systems to thermosyphons. Bezrodnyi (1978) proposed a correlation similar to Kutateladze’s with the Kutateladze number being determined by the vapor pressure and other properties for thermosyphons. Tien and Chung (1978) combined the Kutateladze and the Wallis correlations to account for the diameter of the pipe and surface tension effects. This correlation resulted in agreement with experimental data for certain types of working fluids, but large deviations were found when water was used.

Faghri et al. (1989) improved the existing semi-empirical correlations to predict the flooding limit for thermosyphons by including the effect of diameter, surface tension, and working fluid properties. This is the most general flooding correlation in existence for thermosyphons, and therefore a detailed discussion is presented here.

The Wallis correlation (1969), in which experimental data were correlated for packed beds and countercurrent flow in tubes, is represented by the following empirical equation

${{\left( j_{v}^{} \right)}^{{1}/{2}\;}}+m{{\left( j_{\ell }^{} \right)}^{{1}/{2}\;}}={{C}_{w}}$	(1)

where $j_{i}^{*}={{j}_{i}}\rho _{i}^{{1}/{2}\;}{{\left[ gD\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right) \right]}^{-{1}/{2}\;}}$ ,

$\left( i=\ell ,v \right)$	(2)

in which m and C_w are empirically determined dimensionless constants and are functions of fluid properties. The quantity C_w also depends on entrance and exit geometries. j_l and j_v are the liquid and vapor volumetric flow rates divided by the total cross-sectional area of the thermosyphon. These volumetric fluxes represent ratios of the momentum fluxes of the components to the buoyant forces. Values of m and c were traditionally determined from graphs of $j{{_{v}^{*}}^{{1}/{2}\;}}$ as a function of $j{{_{\ell }^{*}}^{{1}/{2}\;}}$ for open systems. For most cases, values of m = 1.0 and C_w = 0.7 and 1.0 are reported in the literature. Wallis suggested that the liquid film would always flow upward if $j_{v}^{*}>1$ and would flow downward, wetting a dry wall below it, if $j_{v}^{*}<0.5$ . Thus, $j_{v}^{*}=1$ is considered to be the case of total flooding $\left( j_{\ell }^{*}=0 \right )$ . For thermosyphons, as opposed to open systems, $j_{v}^{*}$ and $j_{\ell }^{*}$ are related due to the fact that it is a closed system under steady state conditions.

${{j}_{v}}=\frac{{{{\dot{m}}}_{v}}}{{{\rho }_{v}}A}$	(3)

${{j}_{\ell }}=\frac{{{{\dot{m}}}_{\ell }}}{{{\rho }_{\ell }}A}$	(4)

where ${{\dot{m}}_{v}}={{\dot{m}}_{\ell }}={q}/{{{h}_{\ell v}}}\;$ , A is the total cross-sectional area of the thermosyphon and q is the heat rate. Combining eqs. (1) – (4) and rearranging will result in an equation for the flooding limit based on an extension of the Wallis correlation for open systems to thermosyphons.

$\frac{q}{A}=\frac{C_{w}^{2}{{h}_{\ell v}}\sqrt{gD\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right){{\rho }_{v}}}}{{{\left[ 1+{{\left( {{{\rho }_{v}}}/{{{\rho }_{\ell }}}\; \right)}^{{1}/{4}\;}} \right]}^{2}}}$	(5)

Tien and Chung (1978) extended the Kutateladze correlation for cases with ${{j}_{\ell }}=0$ by an analogy of the Wallis correlation. According to the Kutateladze correlation

${{\left( {{K}_{v}} \right)}^{{1}/{2}\;}}+{{\left( {{K}_{\ell }} \right)}^{{1}/{2}\;}}={{C}_{k}}$	(6)

where ${{C}_{k}}=\sqrt{3.2}$ and ${{K}_{i}}={{j}_{i}}\rho _{i}^{{1}/{2}\;}{{\left[ g\sigma \left( {{\rho }_{\ell }}-{{\rho }_{v}} \right) \right]}^{-{1}/{4}\;}}$ ,

$\left( i=\ell ,v \right)$	(7)

The dimensionless group K (Kutateladze number) is a balance between the dynamic head, surface tension, and gravitational force. Comparing the Kutateladze correlation, eq. (7), with the Wallis correlation, eq. (1), the following relation is found assuming the two are identical.

$D={{\left( \frac{{{C}_{k}}}{{{C}_{w}}} \right)}^{4}}\sqrt{\frac{\sigma }{g\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)}}$	(8)

The critical wavelength of the Taylor instability is:

${{\lambda }_{\text{crit}}}=\left( 2\pi \sim 2\pi \sqrt{3} \right)\sqrt{\frac{\sigma }{g\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)}}$	(9)

Setting eq. (9) (characteristic length) equal to eq. (8) and choosing the upper limit in eq. (8) results in

$\frac{{{C}_{k}}}{{{C}_{w}}}=\sqrt{3.2}$	(10)

If we let C_w = 1.0, eq. (10) gives ${{C}_{k}}=\sqrt{3.2}$ . $K=C_{k}^{2}=3.2$ is really the Kutateladze number for total flooding ( ${{K}_{\ell }}=0$ ), which does not consider the effect of diameter as mentioned in the preceding section.

According to the experimental results of Wallis and Makkenchery (1974), the Kutateldaze number decreases as the dimensionless diameter decreases, which is called the Bond number. This trend requires that C_k is in terms of the Bond number

$\text{Bo}=D={{\left( \frac{{{C}_{k}}}{{{C}_{w}}} \right)}^{4}}{{\left[ \frac{\sigma }{g\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)} \right]}^{{1}/{2}\;}}$	(11)

$\frac{{{C}_{k}}}{{{C}_{w}}}=\text{B}{{\text{o}}^{{1}/{4}\;}}$	(12)

With reference to the variation of the Kutateladze number versus the Bond number in the paper by Wallis and Makkenchery (1974) with $j_{v}^{*}=1.0$ , the function $y = tanh x$ is introduced to account for the effect of the diameter on the flooding limit. If we let $x=\text{B}{{\text{o}}^{{1}/{4}\;}}$ , eq. (12) results in $\frac{{{C}_{k}}}{{{C}_{w}}}=\sqrt{3.2}\tanh \text{B}{{\text{o}}^{{1}/{4}\;}}$ , with C_w = 1.0 or

${{\left( {{K}_{v}} \right)}^{{1}/{2}\;}}+{{\left( {{K}_{\ell }} \right)}^{{1}/{2}\;}}=\sqrt{3.2}\tanh \text{B}{{\text{o}}^{{\text{1}}/{4}\;}}={{C}_{k}}$	(13)

This correlation was found to be highly accurate in predicting the flooding limit with water as the working fluid. The experimental deviations from this correlation are with 15%. For other fluids the deviations are more significant. It is noticed that for different working fluids, the variation of the ratio of the density of the liquid to the density of the vapor is quite different within the same temperature range. Using experimental results for different working fluids, the following correlation is proposed

$\text{K}=C_{k}^{2}={{\left( \frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right)}^{0.14}}{{\tanh }^{2}}\text{B}{{\text{o}}^{{\text{1}}/{4}\;}}={R}'{{\tanh }^{2}}\text{B}{{\text{o}}^{{\text{1}}/{4}\;}}$	(14)

or, for the maximum heat transfer rate,

${{q}_{\max }}=\text{K}{{h}_{\ell v}}A{{\left[ g\sigma \left( {{\rho }_{\ell }}-{{\rho }_{v}} \right) \right]}^{{1}/{4}\;}}{{\left[ \rho _{v}^{-{1}/{4}\;}+\rho _{\ell }^{-{1}/{4}\;} \right]}^{-2}}$	(15)

Equation (15) (Faghri et al., 1989) is a combination of the correlations developed by Tien and Chung (1978) and Imura et al. (1983). In the Tien and Chung correlation, the argument of the hyperbolic tangent in eq. (13) is 0.5Bo^1/4, rather than Bo^1/4, and $R' = 1$ .

References

Bezrodyni, M.K., 1978, “The Upper Limit of Maximum Heat Transfer Capacity of Evaporative Thermosyphons,” Teploenergetika, Vol. 25, pp. 63-66.

Faghri, A., Chen, M. M., and Morgan, M., 1989, “Heat Transfer Characteristics in Two-Phase Closed Conventional and Concentric Annular Thermosyphons,” ASME Journal of Heat Transfer, Vol. 111, No. 3, pp. 611-618.

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

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Imura, H., Sasaguchi, K., and Kozai, H., 1983, “Critical Heat Flux in a Closed Two-Phase Thermosyphon,” International Journal of Heat and Mass Transfer, Vol. 26, pp. 1181-1188.

Kutateladze, S.S., 1972, “Elements of Hydrodynamics of Gas-Liquid Systems,” Fluid Mechanics – Soviet Research, Vol. 1, pp. 29-50.

Tien, C.L., and Chung, K.S., 1978, “Entrainment Limits in Heat Pipes,” Proceedings of the 3rd International Heat Pipe Conference, Palo Alto, California, pp. 36-40.

Wallis, G., 1969, One-Dimensional Two-Phase Flow, McGraw-Hill, New York, NY.

Wallis, G.B., and Makkenchery, S., 1974, “The Hanging Film Phenomenon in Vertical Annular Two-Phase Flow,” ASME Journal of Fluids Engineering, Vol. 96, pp. 297-298.

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Flooding or entrainment limit

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