Cavitation and Gas Dissolution

When the static pressure at a particular location within a liquid falls below the saturation vapor pressure of the liquid, the liquid undergoes a phase change called cavitation. This phase change creates cavitation bubbles that are filled with the vapor of the liquid. Liquid can dissolve a gas up to a certain maximal concentration (saturation). When there are changes in the pressure or temperature (or both), the liquid can become over-saturated or under-saturated. In the over-saturated case, gas bubbles will form and grow in the liquid; in the under-saturated case the gas bubbles will shrink and disappear. For a multi-component gas phase, the bubbles of free gas are a mixture of those components that come out of the solution. The presence of dissolved and free gas in a liquid affects other properties, such as the rates of cavitation.

Simcenter STAR-CCM+ uses the same homogeneous seed-based approach for both cavitation and gas dissolution models. Although these zero-dimensional approaches do not capture all of the physical phenomena present, they are proven methods for use in industrial design studies.

Seed-based mass transfer models are used for gas dissolution and cavitation. The seed-based models are based on [604]. The two interacting phases are denoted as l and v, with mass transfer positive from l to v.

An indicator for the likelihood of cavitation occurrence is the cavitation number $N_{c a v}$ . This value is the difference between the static pressure $p$ and the saturation pressure $p_{s a t}$ , divided by the dynamic pressure of the incoming flow:

Figure 1. EQUATION_DISPLAY

N_{c a v} = \frac{p - p_{s a t}}{{\frac{1}{2} ρ_{l} U}^{2}}

(2679)

In order for cavitation to occur, the cavitation bubbles generally need a surface on which to nucleate. Impurities in the liquid can provide this surface. These pre-existing bubbles are called seeds and start to grow when the conditions for cavitation are met.

In order to be able to account for the effects of liquid purity (that is, the content of solid particles and dissolved gas), it is assumed that nucleation sites (seeds) are present in liquid. The following modeling assumptions are made:

Seeds are spherical and uniformly distributed in liquid, as characterized by the number $n_{0}$ (the number of seeds per unit volume of liquid).
All seeds initially have the same radius.

It is assumed that the number of seeds in a control volume is proportional to the amount of liquid, so there is no need to follow each seed and track its motion in space.

Under the above assumptions, the number of seeds within a control volume at any time is:

Figure 2. EQUATION_DISPLAY

N = n_{0} α_{l} V

(2680)

where $α_{l}$ is the volume fraction of liquid within a control volume of volume $V$ .

The total vapor volume inside the control volume equals:

Figure 3. EQUATION_DISPLAY

V_{v} = N V_{b}

(2681)

where $V_{b}$ is the volume of one bubble, and:

Figure 4. EQUATION_DISPLAY

V_{b} = \frac{4}{3} π R^{3}

(2682)

with $R$ being the local bubble radius.

The vapor volume fraction $α_{v}$ can now be expressed in terms of $n_{0}$ , $R$ , and volume fraction of liquid, $α_{l}$ , following the definition:

Figure 5. EQUATION_DISPLAY

α_{v} = \frac{V_{v}}{V} = \frac{N V_{b}}{V} = \frac{4}{3} π R^{3} n_{0} α_{l}

(2683)

When the volume fraction of vapor is known, the local radius of bubbles can be computed from this expression:

Figure 6. EQUATION_DISPLAY

R = {(\frac{3 α_{v}}{4 π n_{0} α_{l}})}^{1 / 3}

(2684)

The modeling framework that is outlined above is used to define the rate of vapor production or consumption in the mass conservation equation. The rate of bubble radius change is the key parameter, as vapor production results in bubble growth and vapor consumption results in bubble collapse. Another parameter is the amount of liquid in the control volume. Obviously, any increase of the vapor mass is accompanied by a corresponding decrease in the liquid mass.

The vapor bubbles are moving with the flow. Therefore, the rate at which vapor is created at any particular time is approximately the rate at which the volume of bubbles present in the control volume at that time is changing. The change of volume of an individual bubble is:

Figure 7. EQUATION_DISPLAY

\frac{{d V}_{b}}{d t} = 4 π R^{2} \frac{d R}{d t} = 4 π R^{2} v_{r}

(2685)

where $v_{r} = d R / d t$ denotes the bubble growth velocity.

The rate of growth (or collapse) of all bubbles present in a control volume represents the source term in the equation for vapor volume fraction, that is:

Figure 8. EQUATION_DISPLAY

Q_{v} = N \frac{d V_{b}}{d t} = 4 π n_{0} (1 - α_{v}) V R^{2} v_{r}

(2686)

The above source term represents the contribution from cavitation only; vapor can also be produced by boiling. For example, the same flow can have vapor generation at a hot wall due to boiling and vapor generation at a sharp corner due to locally low pressure and cavitation. In such cases, the source terms that result from different models must be added up.

The volume fraction of the vapour can be expressed as:

Figure 9. EQUATION_DISPLAY

α_{v} = \frac{V_{v}}{V} = \frac{V_{v}}{V_{v} + V_{l}}

(2687)

where $V$ is the control volume, and $V_{v}$ and $V_{l}$ denote the respective parts of the volume that the vapour and the liquid phases occupy.

Within the control volume, the vapour phase is assumed to be present in the form of bubbles. Each bubble has the same radius R. The number density of seeds is defined as $n_{0}$ , which corresponds to the number of bubbles per unit volume.

Therefore the following relationship holds between the vapour and liquid phases:

Figure 10. EQUATION_DISPLAY

V_{v} = n_{0} V_{l} \frac{4}{3} π R^{3}

(2688)

The volume fraction of gas, $α_{v}$ , becomes:

Figure 11. EQUATION_DISPLAY

α_{v} = \frac{n_{0} \frac{4}{3} π R^{3}}{1 + n_{0} \frac{4}{3} π R^{3}}

(2689)

The vapour bubble radius can be expressed as:

Figure 12. EQUATION_DISPLAY

R^{3} = \frac{α_{v}}{n_{0} \frac{4}{3} π (1 - α_{v})}

(2690)

The seed diameter $D_{0} = 2 R_{0}$ (which is user-specified) provides a lower limit $R_{m i n}$ for the bubble size.

The mass transfer rate per unit volume can be written as:

Figure 13. EQUATION_DISPLAY

\dot{m} = n_{0} α_{l} 4 π ρ_{v} R^{2} v_{r}

(2691)

For multi-component cavitation, the overall cavitation rate is broken down into rates for the individual components. Unresolved interfacial properties also affect this overall rate. A reasonable assumption is to use the following:

Figure 14. EQUATION_DISPLAY

{\dot{m}}_{c a v, i} = {\dot{m}}_{c a v} ((X_{i, l, \infty} {p^{*}}_{s a t, i}) / (\sum_{i}^{N_{c}} X_{i, l, \infty} {p^{*}}_{s a t, i}))

(2692)

This expression represents a weighting that is based on the individual fugacity of each component, in the context of ideal gases and ideal liquids.

The liquid volume disappears with a volume rate:

Figure 15. EQUATION_DISPLAY

{\dot{V}}_{l} = - \frac{\dot{m}}{ρ_{l}}

(2693)

Vapour volume is generated with a volume rate:

Figure 16. EQUATION_DISPLAY

{\dot{V}}_{g} = \frac{\dot{m}}{ρ_{v}}

(2694)

The remaining unknown factor is $v_{r}$ giving the rate at which the bubble radius changes.

The following models for the bubble radius change rate are provided:

The Rayleigh–Plesset cavitation model
The Schnerr-Sauer cavitation model
The Gas Dissolution model

If the phases are multi-component, the rate model breaks down the global mass transfer rate $\dot{m}$ in rates ${\dot{m}}_{i}$ for each individual component. These component mass transfer rates appear as source terms in the species transport equations.

Full Rayleigh–Plesset Cavitation Model

The Full Rayleigh–Plesset model includes the influence of bubble growth acceleration, as well as viscous and surface tension effects.

The bubble growth velocity $v_{r}$ is determined using the Rayleigh-Plesset equation:

Figure 17. EQUATION_DISPLAY

R \frac{d v_{r}}{d t} + \frac{3}{2} v_{r}^{2} = \frac{p_{s a t} - p}{ρ_{l}} - \frac{2 σ}{ρ_{l} R} - 4 \frac{μ_{l}}{ρ_{l} R} v_{r}

(2695)

where:

$p_{s a t}$ is the saturation pressure for given temperature
$p$ is the local pressure in the surrounding liquid
$σ$ is the surface tension
$ρ_{l}$ is the liquid density

The bubble-growth velocity that is needed for the source term of the vapor volume fraction equation can be computed without neglecting any of the terms in the Rayleigh-Plesset equation.

For multi-component cavitation, the term $p_{s a t}$ in the formulation is replaced with $p_{s a t, m}$ .

Figure 18. EQUATION_DISPLAY

p_{s a t, m} = \sum_{i}^{N_{c}} X_{i, l, s} {p^{*}}_{s a t, i}

(2696)

This expression is Raoult’s Law, with ${p^{*}}_{s a t, i}$ indicating the saturation pressure of the pure component. The interfacial molar fraction of the liquid components is approximately their value in the bulk, so $X_{i, l, s} \approx X_{i, l, \infty}$ . This formulation neglects the effect of other gases that are present in the bubbles (for example, as a consequence of gas dissociation).

Schnerr-Sauer Cavitation Model

The Schnerr-Sauer cavitation model is based on a reduced Rayleigh–Plesset (RP) equation and neglects the influence of bubble growth acceleration, viscous effects, and surface tension effects. The Schnerr-Sauer cavitation model lets you scale the bubble growth rate and collapse rate for both single-component materials and multi-component materials.

The cavitation bubble growth rate is estimated using the inertia controlled growth model [604]:

Figure 19. EQUATION_DISPLAY

{v_{r}}^{2} = \frac{2}{3} (\frac{p_{s a t} - p}{ρ_{l}})

(2697)

where $p_{s a t}$ is the saturation pressure corresponding to the temperature at the bubble surface, $p$ is the pressure of the surrounding liquid and $ρ_{l}$ is the liquid density. Eqn. (2697) is a simplification of the more general Rayleigh-Plesset equation which accounts for the inertia, viscous effects, and surface tension effects.

Research results suggest that for most practical applications, it is not necessary to account for the viscous and surface tension effects [581].

For multi-component cavitation, the term $p_{s a t}$ in the formulation is replaced with $p_{s a t, m}$ .

Gas Dissolution Model

For single-component bubbles, the approximated Epstein-Plesset [588], [596], [599] formulation is assumed to hold:

Figure 20. EQUATION_DISPLAY

v_{r} = \frac{D_{i, l} (c_{i, l, \infty} - c_{i, l, s})}{ρ_{g} R}

(2698)

Due to the small fraction of $c_{i, l}$ present, this formulation can be written as:

Figure 21. EQUATION_DISPLAY

v_{r} = \frac{D_{i, l} ρ_{l} (Y_{i, l, \infty} - Y_{i, l, s})}{ρ_{g} R}

(2699)

To determine the saturation mass fraction, Henry’s law is used, in the following form:

Figure 22. EQUATION_DISPLAY

X_{i, g, s} = \frac{H_{i}}{p^{a b s}} X_{i, l, s}

(2700)

Figure 23. EQUATION_DISPLAY

X_{i, l, s} = \frac{p^{a b s}}{H_{i}} X_{i, g, s}

(2701)

where $H_{i}$ is the Henry’s law constant.

This expression is similar to Raoult’s law, but is valid in the limit of $X_{i, l, s} \to 0$ , whereas Raoult’s law holds in the opposite limit.

In single-component gas dissolution, $X_{i, g, s} = 1$ .

To convert molar fraction to mass fraction, use the assumption:

Figure 24. EQUATION_DISPLAY

\sum_{j}^{N} X_{j, l} W_{j} \approx \sum_{j}^{N_{s v}} X_{j, l} W_{j}

(2702)

with:

Figure 25. EQUATION_DISPLAY

\sum_{j}^{N_{s v}} X_{j} \approx 1

(2703)

where $N_{s v}$ is the number of solvent components. The dissolved species barely contribute to the molar mass of the liquid mixture, due to their low concentration. Therefore, a good approximation is:

Figure 26. EQUATION_DISPLAY

Y_{i, l, s} \approx (X_{i, l, s} W_{i}) / (\sum_{j}^{N} X_{j, l, \infty} W_{j})

(2704)

Eqn. (2699) determines the bubble radius change rate. In the single-component case, the partial pressure of this component is equal to the total gas pressure. The extension to multi-component is readily achieved by assuming ideal gas behavior. Each component contributes to the pressure according to its molar fraction in the free gas, and the species distribution within the bubble maintains a constant profile. In that case, $X_{i, g, s}$ becomes a transported quantity that is influenced by the gas dissolution rates.

The bubble radius rate change is computed as:

Figure 27. EQUATION_DISPLAY

\frac{d R}{d t} = \sum_{i}^{N_{s l}} \frac{D_{i, l} ρ_{l} (Y_{i, l, \infty} - Y_{i, l, s})}{ρ_{g} R} = \sum_{i}^{N_{s l}} {(\frac{d R}{d t})}_{i}

(2705)