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Nodal mass, enthalpy and boron balances

Fig 1 represents the node and its connections.

For Eulerian balances, the control volume is the node geometrical volume v.
For the purpose of setting up the balance relations, the node can be viewed as a perfect mixing cell whose content remains homogeneous.

The inlet "e" [Entry] actually collects the flows of all the inlet junctions whose value (positive or not) are assumed.
The outlet "s" [Sortie] represents the exit junctions whose total flow is to be calculated from the conservation equations.
The flows ws, whs, wbs, as well as the heat rate q received by the node are δs (time step) -mean values.
(s stands here as shorter notation for the current time sec at eos, in order to avoid confusion with the fluid temperature t).

The general constitutive relations (2:5) of the (m,h,b)-balances are classical.
The eos ("end of step") m2, h2, b2, as well the δs-mean value of outlet flows ws, whs, wbl unknowns are collected on the left side of the equations. (008\equations_08.html).
In conformity with our simplifying assumptions [], the kinetic and gravity contributions are absent in the enthalpy balance (3).

Heat conduction model for calculation of q.
In (1), u represents here the spatial average of the conductor temperature, as well as its spatial distribution.
The calculation scheme [] for q is "u-implicit", which means that the heat rate is calculated from the eos temperature u2, but generally t-explicit (calculated from bos water temp t1) because t2 will only be available (7) after that the h-balance (6) is completed .
This t-explicit scheme is acceptable for handling heat-exchange with fuel rods, metallic structures in the loops, pressu and SG, but not for the heat-exchange through the SG tubes because a small change of primary node temperature tli causes a large change of the exchanged heat.
Therefore a special tli-implicit model had to be developed for that case.
The energy balance (3) is set up directly in terms of enthalpy. In the pressure correction term (which results from internal energy to enthalpy conversion) p3 denotes the currently available (appearing thus on right side) guess value of the unknown eos pressure p2.
As explained in the introduction [], the contribution
m (cē/2 + g z/vm)
has been dropped from the balance. This, actually, amounts to overheating the fluid by a temperature excess which can be estimated as
thp cē/2, where
thp = (∂t/∂h)p.
For c = 10, 40, 80 m/s, the temperature excess is only .0016, .025, 1.0 K, respectively.
Likewise, the "gravity overheating" (g z / vm) is also converted to energy.
(5) expresses the nodal volume conservation balance.
(6) represents the implicit equation relating water properties at eos.
(7) is, the explicit calculation of water temperature.
(8:10) are actually the definitions of step-averaged outlet values hs and bs from the corresponding fluxes.
Additional information's (11,12) are required as closure relations in order to get a complete system of 12 equations for the 12 unknowns (m2, h2, b2, vm2, t2, q, u2, ws, whs, wbs, hs, bs).
Three models have been implemented in the program for relating (11) the external outlet enthalpy hs and boron bs to the corresponding bos and eos internal values .
A direct eulerian homogeneous model (13) (cf ) valid for homogeneous flow (no vapor-to-liquid drift) with ws > 0.
According to our homogeneous representation of the node, hs should, ideally, be calculated as the mean value, over δs, of the exponential evolution of h (s) from h1 to h2.
However, this would require adjusting the relaxation factor with δs and node size, would incur the risk of instability and would not correct the forward numerical diffusion effect because the node homogeneity is, anyway, a fiction.
In practice, the totally implicit scheme (om2 = 1, om1 = 0) is solely recommended and the semi-implicit or explicit schemes have only been made available for the purpose of numerical experimentation.
An inverse eulerian homogeneous model (15) valid for negative outlet flow.
It applies, for example, to the Outlet node of the pressu in case wos, escaping to the expansion line, is negative (insurge).
In such a case, hs is a known assumed value whilst ws will result from the balance relations (1:10) closure.
It is easy to show that for the inverse model, h2, hs and vm2 are linked by a linear relation (15) where b is calculated (24) from known values.
An eulerian drift direct scheme which applies, for example, in the "core" of the SG.

Resolution algorithm for eulerian homogeneous flow

For totally implicit case (om2=1), h2 and b2 can be directly calculated (16:20).
(21) is solved for vm2.
This allows completing the algorithm (22, 23).
If ws > 0, the assumptions hs = h2 and bs = b2 are confirmed and the calculation terminates; it provides a closed, local and exact solution (no linearization required) of the nodal conservation equations.
In the (rare) cases where ws < 0, (15) and (24) are substituted into (21), which allows solving the water property equation in terms of vm2, and the algorithm terminates (25:29).

The local conservation is automatically insured but the global solution is not necessarily exact and stable because it relies on the downstream node enthalpy.
If instability occurs, it can be controlled by repeating the node balance sweeping to improve the exactness of the global solution.