16.2. In-pin Hydrodynamic Model¶
16.2.1. Physical Models¶
The in-pin hydrodynamic model describes the motion of the molten fuel and fission gas mixture in the cavity formed inside the fuel pins during a loss of flow accident. As the accident proceeds, the size of the cavity increases, both radially and axially (Figure 16.1.2). Newly molten fuel and fission gas are added to the moving components in the cavity. Some of the fission gas is dissolved in the molten fuel, in the form of small bubbles constrained by surface tension. The effect of this fission gas is controlled by the input variable PRSFTN. If PRSFTN is less than \(10^{7}\), the volume of the dissolved gas is assumed to be negligible, and thus it does not contribute immediately to the cavity pressure. If \(\text{PRSFTN} > 10^{7}\), the volume occupied by the dissolved gas is taken into account in the cavity pressure calculation, as described in Section 14.2.6. The remainder of the fission gas is in the form of free gas, residing in bubbles that are too large to be constrained by surface tension. This gas contributes immediately to the pressurization of the cavity. Because of the continuous coalescence of the small bubbles, leading to the formation of new larger bubbles, the originally dissolved gas is continuously released from the molten fuel and is added to the free fission gas. The continuous heating of the molten fuel and fission gas leads to the pressurization of the cavity and eventually to the cladding failure. The fuel and fission gas in the vicinity of the failure location are ejected into the coolant channel leading to a local depressurization of the cavity. This depressurization causes the fuel motion inside the pin toward the failure location.
The in-pin fuel motion is treated as a one-dimensional, compressible flow with a variable flow cross section. The fuel and fission gas are assumed to form a homogeneous mixture in thermal equilibrium. However, if the local fuel volume fraction is less than an input value FNFUAN, the pressure gradient is assumed to act only on the fuel cross section. This is an attempt to roughly account for the annular fuel flow regime which may exist for large void fractions. Fuel vapor pressures are also included in the in-pin hydrodynamic model. The fuel vapor pressure is based on the average fuel temperature in any axial cell, with the assumption that no significant radial temperature profiles can be maintained after the onset of fuel motion.
As the cladding heats up and the in-pin pressures increase, the cladding rip can propagate from the original location, allowing the ejection of fuel and fission gas from the cavity into the channel at new axial locations. The fuel pin can also be totally disrupted at certain axial locations where the cladding becomes very weak and the solid pin is largely molten. The pin disruption leads to the formation of upper and lower pin stubs with cavities that can eject fuel axially into the disrupted region.
16.2.2. Method of Solution and General Numerical Considerations¶
16.2.2.1. Variables and Mesh Grid Used in Calculations¶
The independent variables used in the in-pin model are the axial coordinate \(z\) and the time \(t\). Only one spatial coordinate is necessary, as LEVITATE models the pin cavity in a one-dimensional geometry. The dependent variables calculated by the in-pin hydrodynamic model for each component are the generalized density \({\rho'}\), the enthalpy h (or temperature T), and the velocity u. The generalized densities have been introduced in Chapter 14 and, for the component j in the cavity, are defined as follows:
(16.2-1)
where
\(\rho_{\text{j,ca,k}}\) is the physical density of component \(j\) at the axial location \(k\) in the pin cavity
\(A_{\text{j,ca,k}}\) is the cross sectional area occupied by component \(j\) at location \(k\) in the pin cavity. This area refers to all the pins in the subassembly.
\(\text{AXMX}\) is the reference input area
\(\theta_{\text{j,ca,k}}\) is the generalized volume (or area) fraction of component \(j\) at location \(k\) in the pin cavity
The mass, energy and momentum partial differential equations are solved using the Eulerian finite difference semi-explicit formulation, as explained in Section 16.2.2.2. A staggered mesh grid is used to obtain the numerical formulation, with the densities and enthalpies defined at the center of each cell while the velocities are defined at the boundaries. Variable flow areas are treated in the in-pin model. However, the cavity geometry is not as irregular as the coolant channel geometry, and the conventional single velocities were used at the boundaries, as opposed to the dual velocities used in the coolant channel model (see Section 16.4.2.1). The components treated in the cavity model are the molten fuel, the dissolved fission gas and the free fission gas. They are assumed to form a homogeneous mixture and move with the same velocity at all axial locations. A full donor cell formulation was used in the finite difference formulation in order to improve the stability characteristics of the solution.
16.2.2.2. Description of the Method of Solution and Logic Flow¶
The in-pin hydrodynamic model is solved in the routines LE1PIN and LE2PIN. First, the cavity enlargement is calculated in LE1PIN, using the solid fuel temperatures provided by PLHTR. If melting occurs, the quantities of molten fuel and fission gas to be added to the moving components are determined. Then the mass conservation equation is solved for all axial cells, accounting for the changes in fuel and free fission-gas mass due to convection and melt-in. LE1PIN then solves the energy conservation equations for the fuel and fission-gas mixture. It is assumed that the fuel and fission gas remain at the same temperature in all the axial cells. Using the new masses and temperatures, the new pressures are then calculated. It is noteworthy that at this point the pressures have also been updated in the channel (see Section 16.4.2.2) so that we can use a consistent set of pressures for the ejection calculation, which is the next step in the LE1PIN routine. The ejection calculation leads to changes in the mass and pressure in all nodes that are ejecting fuel and fission gas into the coolant channel. The ejection can take place radially, via the cladding rip or axially, via the open ends of the fuel-pin stubs when the fuel-pin disruption has already occurred. Both modes of ejection can be present simultaneously. The LE1PIN routine also examines the possibility of pin disruption. If an axial cell is found which has to be disrupted, a flag is set (IDISR(I)=9), but no other changes are preformed. The actual pin disruption is performed in the routine LEDISR, as described in Section 16.5.
The routine LE2PIN then solves the momentum conservation equation for all cells, obtaining the new velocities at the end of the time step. These velocities are obtained by using the new pressures calculated in LE1PIN, and in this sense, the method of solution is mixed, explicit-implicit, rather than purely explicit. The routine LE2PIN also solves the mass conservation equations for the dissolved gas and calculates the maximum time step acceptable for the in-pin hydrodynamic model in the next computational cycle.
16.2.3. Finite Difference Forms and Solution Technique, Special Situations¶
The equations describing the in-pin hydrodynamic and thermal process are solved in the LE1PIN and LE2PIN routines. The partial differential equations, as well as the finite difference formulation, are generally the same in LEVITATE and PLUTO2. The reader is referred to the PLUTO2 chapter (Section 14.2) for a detailed description of the equations. Only the features of the in-pin model that are specific to LEVITATE will be discussed here.
The main feature of LEVITATE is the fuel-pin disruption mode. The decision for disruption of a certain axial pin cell is made in the routine LE1PIN. An undisrupted cell will be disrupted if the molten fuel cavity covers a large fraction of the original pin cross section.
(16.2-2)
and the cladding is molten or close to melting, i.e.,
(16.2-3)
and:
(16.2-4)
or:
(16.2-5)
The disruption of one or more axial pin nodes leads to the formation of a disrupted region, extending from the cell IDISBT to IDISTP. Note that the axial position of the in-pin cells is denoted in Eq. 16.2-1 by the subscript \(k\), while the subscript used in the channel is \(i\). The correspondence between \(i\) and \(k\) is given below:
(16.2-6)
that is, the in-pin cell \(k\) will have the same axial location as the channel cell \(k + \text{IDIFF}\).
The motion of the material present in this region is calculated by the coolant channel hydrodynamic model. In the disrupted region, the coolant channel covers the entire cross sectional area of the subassembly. Only one disrupted region is allowed in LEVITATE. Thus, if two or more disjoint disrupted regions appear at any given time (e.g., cells 8 and 10 are disrupted, but not 9), the undisrupted nodes between the regions will also be forced to disrupt. In this case, a message is printed indicating that one or more nodes have been disrupted due to the disruption of neighboring nodes and the formation of large fuel/steel chunks is likely to occur. Only the decision about disruption is made LE1PIN. The nodes to be disrupted are flagged (\(\text{IDISR} \left( I \right) = 9\)), but the disruption process will only be modeled later in the cycle, in the routine LEDISR. This process is described in Section 16.3.
After the occurrence of the fuel-pin disruption, the model describing the hydrodynamics of the molten fuel/fission gas in the pin cavity continues to operate. However, instead of modeling a continuous channel, it now describes the hydrodynamic behavior of two disjoint channels, i.e., the cavities remaining in the upper and lower undisrupted pin stubs. These cavities communicate directly with the channel via the open ends of the pin stubs. The channel pressure in front of these open ends is used as the boundary condition for the in-pin hydrodynamic model.