12.4. Bubble Formation and Collapse

At the end of each time step before voiding starts in a channel, the coolant temperature at each axial node is compared with the corresponding saturation temperature. The code estimates when boiling will start by linearly extrapolating coolant temperatures and saturation temperatures. A switch to the boiling module is made before the formation of the first bubble. The voiding model then takes over the calculation of coolant temperatures and saturation temperatures by applying the liquid slug model described in Section 12.2 to the entire channel. At the end of each time step, the code checks to see whether the boiling criterion has been met at any point along the axial mesh. This criterion requires that the coolant temperature exceed the saturation temperature by a minimum superheat value. This superheat value is selected by the user and is entered through input variable DTS. Experience with the code has shown that a superheat value of about 10 K is appropriate for the initial bubble in the channel. This amount of superheat prevents the bubble from collapsing immediately after formation and appears to produce results that agree well with the experimental results discussed in the introduction to the chapter.

Once the code predicts that the coolant temperature has exceeded the saturation temperature plus the superheat, it begins an iterative process to determine within a small tolerance the time at which the superheat criterion is satisfied exactly. This is done by checking to see by how much the coolant temperature exceeds the sum of the saturation temperature plus the superheat. This temperature excess is used in a linear interpolation to reduce the time step to a value at which the superheat criterion will be met more closely. The code then goes back to the beginning of the time step and repeats the coolant and saturation temperature calculations and again checks the superheat criterion. If the criterion is met to within a built-in tolerance (0.001 K), the code fixes that time as the time of voiding initiation for the channel and forms a vapor bubble at that point in the channel. If the criterion is not met, the code repeats the calculation of the temperature excess beyond the superheat criterion, calculates a new reduced time step, and again goes back to the beginning of the time step and repeats the temperature calculations. This iterative procedure continues until the superheat criterion is met satisfactorily. Normally, the procedure converges with three or four iterations.

After voiding has started in a channel, the voiding model calculation tests at the end of each coolant time step for the formation of new bubbles within the liquid slugs. The formation of new bubbles after voiding has started is subject to the following limitations and modifications.

1. If the maximum number of bubbles (nine bubbles) already exist in the channel, then new bubbles will not be formed.

2. No new bubbles will be formed within a minimum distance, \(s_{Lmin}\) (minimum slug length) of a bubble-liquid interface, so nodes within \(s_{Lmin}\) of an interface are not checked for bubble formation. Interpolated coolant temperatures and saturation temperatures at a distance \(s_{Lmin}\) from each interface are checked, and a bubble is formed at this position if the superheat criterion is exceeded. The minimum distance \(s_{Lmin}\) is an input variable SLMIN; experience with the code indicates that a value of about 2 cm is reasonable for most situations.

3. No more than one new bubble will be formed in a channel in any time step.

4. After the first bubble is formed, time steps are not repeated so as to match exactly the superheat criterion for later bubbles; when the superheat criterion is exceeded, a new bubble is formed with whatever superheat happens to exist. The superheat used in the superheat criterion for bubbles after the first one should be somewhat less than the value used at the start of boiling, generally 3 K or 4 K. This is also a user-input quantity and is read into input variable DTSI. At the end of each time step, linear extrapolation of coolant and saturation temperatures is used to predict when the next bubble will form. This prediction is used to limit the size of future time steps, so that the saturation criterion is normally exceeded only slightly when a new bubble is formed.

Bubbles are assumed to fill the whole cross section of the coolant channel, except for a liquid film left on the cladding and structure. A bubble starts with zero length and with the initial temperature equal to the coolant liquid temperature at the time of bubble formation.

Pressures are continuous across the liquid-vapor interfaces; but sharp temperature gradients can occur in the liquid temperature a short distance from the interface can be significantly different from the vapor temperature at the interface. For each bubble interface, the code calculates one pressure, a vapor temperature, and a liquid temperature at a position far enough from the interface that it is unaffected by the vapor temperature. Also, the bubble interface position and velocity are calculated. In addition to bubble interface values, coolant temperatures, pressures, densities, and flow rate are calculated at the mesh segment interfaces. The liquid film thickness in a voided region is calculated as the average for the segment. Flow areas and hydraulic diameters are treated as constant within a segment although they can vary from segment to segment.

The vapor bubble pressure is assumed to be always equal to the saturation pressure corresponding to its temperature. Thus, the formation of a vapor bubble with a nonzero amount of superheat leads to an immediate jump in pressure at the bubble location.

The initial bubble growth is due mainly to two effects: (1) the initial jump in pressure corresponding to the superheat, drives the liquid slugs apart, forming a larger bubble; and (2) heat flow through the liquid-vapor interface produces vapor to fill the bubble and sustain the pressure. When the bubble gets larger, vaporization of the liquid film on the cladding becomes the main source of vapor. The vapor pressure and temperature are assumed to be spatially uniform during the initial bubble growth. This model for the initial bubble growth is similar to the mode of Fauske and Cronenberg [12-7, 12-8] and also similar to that of Schelechtendahl [12-9].

After the bubble length exceeds a user-specified minimum length (usually 3-10 cm), the vapor bubble calculation is switched to a different model. Heat flow through the liquid-vapor interfaces is ignored, but axial variations in vapor pressure within the bubble are accounted for. Vaporization of the liquid films in the hot part of the core, and condensation in cooler regions, can lead to high vapor velocities and corresponding appreciable pressure gradients within a vapor bubble. The numerical scheme that is used to calculate the time and space variations in the vapor pressure is an implicit scheme that can handle moderately large (a few milliseconds) time steps. Wave effects tend to be suppressed by the numerical scheme.

Condensation in cooler regions can cause a vapor bubble to shrink. If the bubble size decreased to below the minimum length for the pressure variation model, then the initial uniform pressure model is used again. If the bubble size decreases to less than a minimum bubble size, and if at the same time the rate of decrease of the bubble size exceeds a minimum value, then the bubble is assumed to collapse and the bubble is eliminated from the calculation. When a bubble collapse, the liquid slugs above and below the bubble are combined into one slug, and the flow rate of the combined slug is determined by conserving the combined momentum of the two slugs.