As discussed early, it is impossible to have a universal interphase mass transfer model due to the diversity of underlying physical mechanisms. The focus of this work is only on thermodynamic interfacial mass transfers in liquid-gas multiphase flows.

In essence, all the thermodynamic phase mass-exchanges between a gas and liquid phase are either evaporation or condensation. In a liquid-gas two-phase flow, assuming and are the volumetric rate of evaporation and condensation respectively, and ignoring the external mass sources, we rewrite the liquid phase volume fraction transport equation as:

 

5.295

And and have the general expressions:

 

5.296

where represents the interfacial mass transfer source from liquid-to-interface (evaporation); and is the interfacial mass exchange from vapor-to-interphase (condensation). They are determined by a transfer coefficient (heat, mass) and gradient (temperature, mass concentration, or pressure).

In equation 5.296, the variable is the interfacial area concentration. In homogeneous gas-liquid two phase flows, it is estimated using an algebraic model, while in VOF model, it is computed using the volume fraction gradient at the interface.

Clearly, to obtain the volumetric evaporation and condensation rates and , two sets of models are required: the interfacial area concentration (), and the mass transfer rates . The corresponding sub-models are to be discussed in the flowing sections.

Interfacial area concentration is defined as the interfacial area between two phases per unit mixture volume. This is an important parameter for predicting mass, momentum and energy transfers through the interface between the phases. When using the VOF and Mixture multiphase model with non-granular secondary phases, Simerics-MP computes the interfacial area concentration as follows:

Mixture Multiphase Model

The algebraic interfacial area models are derived from the surface area to volume ratio, , for a spherical bubble or droplet with a diameter, :

5.297

The algebraic models available when using the Eulerian multiphase model are:

Particle Model

5.298

Symmetry Model

5.299

where and are vapor and liquid volume fraction, respectively; is the dispersed phase volume-fraction; and is the diameter of the dispersed phase. For bubbly flows (boiling, cavitation), it is the vapor bubble diameter. For mist/ droplet flows (condensation), it is the droplet diameter.

 

Free Surface Model

In the free surface model, the interfacial area concentration is estimated using the magnitude of the phase volume fraction gradients. For two-phase and - phase ( > 2) VOF flows, the interfacial area between phase- and phase- is:

 

Two-Phase Flow

5.300

n-Phase Flow

5.301

 

In multiphase flow and heat transfer, the liquid-vapor evaporation and condensation can be categorized into two groups: volumetric and surface thermal phase change. The volumetric mass transfer occurs between the liquid-vapor interface where the heat is in the state of non-equilibrium. The CFD models for this type of interfacial thermal phase changes have been described as Interfacial Mass Transfer.

The surface thermal phase changes may happen on a wall surface or in a wall boundary layer of a liquid, vapor (steam) or liquid-vapor flow due to the wall-to-fluid (phases) heat transfer. For example, when a sub cooled liquid flow is subjected to a heated surface, wall boiling/ evaporation starts if the wall temperature is sufficiently high to initiate the activation of wall nucleation sites. This activation temperature is typically a few degrees above the saturation temperature. However, at this stage, the average temperature of the liquid in the vicinity of the heated wall is still well below the saturation temperature, hence in the sub-cooled boiling regime.

In Simerics-MP, the present volumetric thermal phase change model considers both interfacial evaporation and condensation, while the surface phase change model only includes wall evaporation (boiling). Note that the interphase volumetric mass transfer can occur independent of wall mass transfers.

Interfacial Mass Transfer

In liquid-vapor two phase flows, the thermal phase change models consider the interfacial mass transfer: the evaporation of the super-heated liquid, and the condensation of the sub-cooled vapor. The fundamental assumption is that when the liquid and vapor are in contact, there is a tendency to attain the local thermal equilibrium (at steady-state): pure liquid with temperature below the saturation temperature, the liquid-vapor mixture at the saturation temperature, and pure vapor with temperature above the saturation temperature. When the local thermal equilibrium is broken, heat and mass transfer could occur, and a new dynamic equilibrium state would be ultimately established.

Assuming that at the interface between the and  phase, “”, the temperature, , is the same on both sides, then the volumetric rates of phase heat exchange can be expressed as follows:

From the phase to interface

5.302

From the phase to interface

5.303

where and are the heat transfer coefficients from the two phases to the interphase “” respectively. With mass transfers, the interphase temperature is the saturation temperature, .

The terms and represent interfacial values of enthalpy carried into and out of the phases due to phase change. They need to be computed correctly to take into account, the discontinuity in static enthalpy due to latent heat between the two phases as well as the heat transfer from either phase to the phase interface. With the bulk fluid enthalpy carried out of the outgoing phase and the saturation enthalpy carried into the incoming phase we have:

If (evaporation, the liquid phase is the outgoing phase):

5.304

If (condensation, the liquid phase is the incoming phase):

5.305

Since neither heat nor mass can be stored on a phase interphase, the overall heat balance must be satisfied:

5.306

From equation 5.302, equation 5.303 and equation 5.306, the rate of interfacial mass transfer can be determined:

5.307

For homogeneous heat transfer multiphase flows, it is assumed that all the phases share the same temperature. For the liquid-vapor two phase flows, using subscripts “” and “” to indicate liquid and vapor phase respectively, and assuming , we have the liquid-to-vapor mass transfer:

5.308

As for the fluid / phase specified heat transfer coefficients in equation 5.308, and , they can be computed using empirical correlations.

 

Ranz-Marshall Model

In Simerics-MP, the Ranz-Marshall correlation1W.E. Ranz and W.R. Marshall Jr, "Evaporation from Drops, Part I and Part II", Chem. Eng. Prog. 48(3):141-146, and 48(4):173-180, 1952. is the default model used to estimate both and :

	5.309
	5.310

where the subscripts “” and “” indicate the dispersed and primary phase (for boiling: is vapor, is liquid; for condensation: is liquid, is vapor); is the primary phase thermal conductivity; is the dispersed phase diameter (bubble diameter in boiling, and droplet diameter in condensation); and is the dispersed phase Nusselt number.

Note that the Ranz-Marshall correlation is in principle, more appropriate for liquid-to-vapor heat transfer. In equation 5.309 and equation 5.310, the same heat transfer coefficients are used on both interfacial evaporation and condensation. In reality, different heat transfer models should be used for modeling the two physical processes. This can be achieved by applying the Ranz-Marshall correlation on each phase separately or choosing entirely different models. In fact, in addition to the Ranz-Marshall formulation, a number of correlations are routinely used to estimate the interfacial heat transfer coefficients.

Constant Heat Transfer Coefficient

The volumetric heat transfer coefficient is determined by a user-specified value on one or all the phases.

Constant Nusselt Number

The Nusselt number is determined by a user-specified value on one or all the phases. For phase-, the volumetric heat transfer coefficient   is given by:

5.311

Hughmark Model

Hughmark correlation 2G. A. Hughmark, “Mass and Heat Transfer from Rigid Spheres”, AICHE Journal, 13. 1219-1221. November 1967. is an extended version of Ranz-Marshall model for a wider range of Reynolds numbers:

5.312

Tomiyama Model

Tomiyama proposed a slightly different correlation to the Ranz-Marshall model for the interfacial heat transfer, applicable to turbulent bubbly flows under relatively lower Reynolds number. It has the expression3A. Tomiyama, "Struggle with Computational Bubble Dynamics". Third International Conference on Multiphase Flow, Lyon, France, June 8–12, 1998.:

5.313

Lavieville et al Model

The Lavieville et al4J. Lavieville, E. Quemerais, S. Mimouni, M. Boucker and N. Mechitoua, "NEPTUNE CFD V1.0 Theory Manual", EDF, 2005. first proposed the “constant time scale return to saturation” method to calculate the vapor-to-interphase heat transfer in wall boiling models. It has then been generalized for computing the gas-interphase heat transfer coefficient. In this model, it assumes that the gas-phase retains the saturation temperature by rapid evaporation or condensation. The formulation is as follows:

5.314

is the time scale and is usually set to 0.05 s . is the gas phase isobaric heat capacity.

Finally, one may note that it is also possible to specify an infinite heat transfer coefficient on a phase. For example, if , its effect is to force the interphase temperature to be the same as the vapor phase temperature, .

 

Wall Boiling-RPI Nucleate Wall Boiling Model

Wall boiling starts in the microscopic cavities and crevices, which are always present on solid surfaces. Liquid becomes supersaturated locally in these nucleation sites, leading to the growth of vapor bubbles at the sites. The bubbles become detached from the sites when they are sufficiently large that external forces (inertial, gravitational, or turbulent) exceed the surface tension forces that keep them attached to the wall. As the bubbles depart from the wall, they are displaced by superheated liquid in the vicinity of the nucleation sites, after which the nucleation site is free to create another bubble. In regions of the wall not affected by bubble growth, the wall heat transfer to the liquid may be described by single phase convective heat transfer.

The first and most well-known wall boiling model to the CFD community is the multi-dimensional, multi-fluid RPI model developed by Kurul & Podowski5N. Kuruland M. Z. Podowski, "On the Modeling of Multidimensional Effects in Boiling channels", In Proceedings of the 27th National Heat Transfer Conference, Minneapolis, Minnesota, USA. 1991.. In this mechanistic model, the total heat flux from a wall to liquid is partitioned into three components: liquid phase convective heat flux, , quenching heat flux, and evaporation heat flux, :

5.315

Assuming that the heated wall surface is subdivided into a portion, covered by nucleating bubbles and the remaining part occupied by fluid, the RPI model gives the following expressions for the three heat flux components:

 

Liquid Phase Convective Heat Flux:

5.316

where is the liquid phase heat transfer coefficient, and are the wall and liquid temperature in the vicinity of the wall respectively.

Quenching Heat Flux:

This term represents the cyclic averaged transient energy transfer related to liquid filling the wall vicinity after the bubble detachment with a period of . It is expressed as:

5.317

where  and are the heat conductivity and diffusivity in the liquid phase; is the bubble departure frequency; is a coefficient introduced to correct the waiting time between departure of consecutive bubbles, which is typically between 0.9 and 1.0. By default, it is set to 0.9, fixing the waiting time between departures of consecutive bubbles at around 80% of the bubble detachment period:

5.318

In equation 5.317, the term is the cross-section-averaged liquid temperature in the originally correlation for one dimensional thermos-hydraulic analysis. In an unstructured general CFD solver, however, it is impossible to compute such an averaged value. The common approach is to set . But this practice leads to be dependent of the size of the near-wall cells, and thus mesh-dependent solutions. To remedy this, the logarithmic form of the wall function is generally used to estimate at a fixed  value, by default, 250, i.e., .

Evaporation Heat Flux:

5.319

where is the bubble departure diameter, is the active nucleate site density, is the vapor density, and is the latent heat of evaporation.

 

To close equation 5.316- equation 5.319, the following parameters/sub-models are employed:

Area of Influence: It is defined on the basis of the bubble departure diameter and the nucleate site density:

5.320

where the coefficient can be set a constant (by default, it is 4, but may vary between 1.8 and 5), or use Del Valle and Kenning's findings6V. H. Del Valle and D. B. R. Kenning, "Subcooled Flow Boiling at High Heat Flux", International Journal of Heat and Mass Transfer, 28(10), 1907–1920,1985.:

5.321

where the liquid sub-cool temperature ; and are the liquid density and heat capacity, and is the saturation temperature.

Frequency of Bubble Departure: This is usually calculated based on inertia controlled growth7R. Cole, "A Photographic Study of Pool Boiling in the Region of the Critical Heat Flux", AIChE J., 6. 533–542, 1960.:

5.322

where is the magnitude of gravity.

Bubble Departure Diameter: Among a number of correlations, the widely used sub-model has the following form, proposed by Tolubinski and Kostanchuk8V. I. Tolubinski and D. M. Kostanchuk, “Vapor Bubbles Growth Rate and Heat Transfer Intensity at Subcooled Water Boiling”, 4th International Heat Transfer Conference, Paris, France. 1970.:

5.323

Here by default, and .

Nucleate Site Density: This quantity is estimated using the wall superheat, :

5.324

By default, and , from Lemmert and Chawla9M. Lemmert and L. M. Chawla, “Influence of Flow Velocity on Surface Boiling Heat Transfer Coefficient in Heat Transfer in Boiling”, E. Hahne and U. Grigull, Eds., Academic Press and Hemisphere, New York, NY, USA. 1977..

 

Wall Boiling-Departed Nucleate Boiling

In the RPI model, the wall heat flux partitions imply that no wall heat directly applies to the vapor or other non-condensable gases in the vicinity of a heat wall. This assumption is valid in the nucleate wall boiling regime, under which the sub-cooled liquid or bubbly liquid-gas flows are subjected to heat transfer in the vicinity of heated wall surfaces. Quantitatively, if the evaporating liquid has a local volume fraction of 0.8 or above, and the wall temperature is higher than the saturation temperature, the direct wall-to-gases heat transfer can be ignored and the vapor temperature is assumed to be at the saturation temperature. In the homogeneous mixture model, the fluid temperature should not exceed the saturation temperature.

When the wall boiling departed from the nucleate boiling regime such as the non-equilibrium boiling, critical heat flux (CHF) and post dry-out, the volume fraction of vapor is significant (typically ≥0.3), and the flow is no longer in the bubbly flow regime. Under those conditions, the wall-to-vapor interactions (direct heat transfer) must be accounted for and the fluid temperature can be over the saturation temperature at CHF and post dry-out conditions. In addition, the presence of thin liquid films along heated walls also needs to be considered in the wall-to-fluid heat transfer. The wall heat partition is thus modified as follows:

 

5.325

 

Here is the thin film boiling heat flux, is the diffusive heat flux of the vapor phase, and represents heat flux to the non-condensable gas phase - in the system. They are calculated by:

5.326

The convective heat transfer coefficients and are computed based on turbulent wall treatment. The model for the film heat transfer coefficient , can be estimated as10A. Ioilev, M. Samigulin and V. Ustinenko, "Advances in the Modeling of Cladding Heat Transfer and Critical Heat Flux in Boiling Water Reactor Fuel Assemblies", The 12th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-12). Pittsburgh, Pennsylvania. 2007.:

5.327

where is the liquid Prandtl number, and are non-dimensional wall distance and universal temperature profile and is the thin film thickness, which is typically prescribed as around 1e-4 m in wall boiling.

 

The function depends on the local liquid/vapor volume fraction with the same limiting values as the liquid volume fraction. Lavieville et al proposed the following expression in terms of the liquid volume fraction :

5.328

The critical value for the liquid volume fraction is 0.2, and for the vapor phase, it is .

There are also some other functions available to define the wall heat flux partition. To capture the sharp changes at the critical heat flux (CHF) conditions, Ioilev et al adopted the following function in terms of vapor volume fraction:

5.329

where the breakpoints have been set as and .