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Mitigating numerical diffusion by Lagrangian homogeneous model

As explained in the introduction, a special lagrangian balances model had to been developed in view of overcoming the "forward diffusion" effect associated with the Euler balances scheme.

For the moment, this lagrangian solution has only been implemented for the loop_l and down_comer components where the transport effect is deemed important as far the reactor thermal protection is concerned.

Picture 1 illustrates the Lagrangian scheme.
Just as for the Euler scheme, the component is spit into i9 fixed nodes i=1,i9 of constant transverse area each.
Each node can receive some uniformly distributed heat rate qi and some flow wyi, supposed to be injected at node inlet and carrying (for positive injection) enthalpy hyi and boron hyi.
The Lagrangian balances (1:4) are not set up on the fixed "nodes" i, but on a set of volume "elements" j traveling with the fluid.
Thus, for zero drift, the boundary "elemental" we and ws flows are zero, but the elemental volume may change by the effect of fluid expansion and/or transverse flow injection or bleeding.
δs qj is the total heat collected by element j from the nodes crossed by j in the course of its travel during δs, and δs whyj the corresponding total enthalpy.
At initial conditions, the component volume vl is filled with elements of constant volume Δv. Thus, vl contains
INT(vl/Δv) complete elements and a last, incomplete element of volume vl-INT(vl/Δv)*Δv
Likewise, the inlet volume δv injected in the time step is filled with INT(δv/Δv) complete elements and one incomplete element of volume δv - INT(δv/Δv)*Δv.
(5:7) represent the values introduced at inlet
The picture shows the "characteristic curve" of each element j, namely the trajectory j which is the graph versus time of the position swept by the end section of vj.
The picture details how the inlet δv must be refilled after each Δs and how the elements must be renumbered in such a way that the elements injected into the component keep a constant volume (except for the expansion and transverse injection effects)
In practice, Δv should be of the same size as the smallest node.
In order to solve the Lagrange balance equations without iterations, it was necessary to accept some simplifying assumptions:
Each characteristic, which is normally curved by the effects of expansion and transverse injection is approximated, within the Δs by a polygon segment.
In the course of each δs, and solely for the purpose of calculating qj and wyj, the element volume is supposed invariant (v2j is approximated by v1j) but, at the outcome of the element balance, m2j is known and the element volume v2j is updated to m2j vm2j in such a way that the global balances of the component is closed exactly.
The qi are calculated from the bos nodal temperatures t1i.
However, if the step is repeated (redo, loop), t2i is known, and the effective temperature is taken as t1i + omtl (t2i - t1i), where omtl is a entered relaxations factor. It is reminded that the Eulerian scheme dit allow time-implicit calculation.
It is also supposed the elements have been chosen large enough to avoid their vanishing due to transverse mass loss by bleeding.
At the issue of each time-step, each node is refilled by homogenizing its intersection with the set of elements.
This operation is provided for the only purpose of node property editing and updating the temperature t1j used for next evaluation of qi.
The elements themselves are, however, not altered. If needed, the elements properties can also be plotted, but this is not convenient because of the j-renumbering after each time-step.

Component exit flows

Cf picture 2.
The last updated δv represents the volume displaced by the last element j19 outlet boundary and the corresponding mass ms, enthalpy mhs and boron mbs is obtained by collecting the corresponding values for the elements (or part of it) which have left the component outlet. It is of course assumed that, as soon the element leaves that outlet, if does not collect any heat or injection any longer.
The exit properties are thereafter obtained from (10:15).
It has been checked (16:18) that the thus calculated exit properties values are numerically identical (except for computer arithmetic's accuracy) to those obtained from global component balances closure.
The exit values hs and bs are not exactly the same as the h2i and b2i which could be observed in a last thin node i9.
Because of the finite size of the elements, the nodal properties and, to a lesser extent, the exit properties may exhibit some erratic behavior.

These effects are well illustrated on application Xln04_d.dat with a step jump of bel at 10 s and a boron injection in node 13.
It is also observed that boron injection in node 16 is already felt in upstream node 15, which is non-physical but is explained by the fact that node 15 may intersect with an upstream element fed by injection in node 16.
This model anomaly could be avoided, if required, by calculating a "local" hs, bs and ts at exit of node 15, by means of model used for component exit. For the moment, the Lagrange balance model has only been implemented for the components loop_n and reactor vessel down_comer where the transport effects may be relevant for the protection based on temperature monitoring in the loops. If required, it could easily been extended for modeling time delay effects in temperature sampling lines, or in the safety injection line.
Transport effects are considered less important for the core and mixture components like the Bottom. A special 2-phase Lagrange model has been developed for the pressu component .