# EDF

## Mixtures of gases to build complex equations of state

**Mathematics**: Modeling

EDF needs reliable numerical studies of flows to reinforce the safety demonstration of its nuclear production plants. To simulate hypothetical accidents, numerical codes require models able to describe complex physical situations and strongly unsteady phenomenon : a classical example is the loss of coolant accident, in which a breach in the primary pipe is assumed. The reactor water, pressurised at 150 bar, is suddenly put in contact with air at atmospheric pressure, leading to a very sudden water vaporization. In such situations as in many others, temperature and pressure vary a lot and can possibly reach domains which are difficult to describe with a simple equation of state, like the neighbourhood of the critical point for instance.

Current works consist in improving the physical description of the water behavior in the models. Indeed, a classical possibility is to describe the water behavior with a simple equation of state, as the stiffened gas equation. Physically, this law is close to a perfect gas equation, which is modified to take into account the particular behavior of a liquid. This law is defined using a reference point, where all the physical quantities are known (pressure, temperature, enthalpy, thermal capacity…). Near this reference point, the equation of state is realistic, but its accuracy decreases when pressure and temperature are too far from the reference point.

Finally, a stiffened gas equation correctly describes a limited domain in the thermodynamical plane. This domain is too limited for the accidental scenarios which might hypothetically occur in a pressurized-water reactor. Our idea is the following: a complex equation of state, more realistic on a wider physical area, could be built as a mixture of several simple equations of state. The starting point is to consider two stiffened gases. Each one is defined using its own reference point, and they are thus accurate on different domains in the thermodynamical plane. The aim is to then build a mixture of this two laws which fulfills the following constraints: *(i) *The mixture law does not decrease the accuracy near the reference point of each stiffened gas, *(ii) *The mixture law is accurate on a wider area than the gathering of accuracy area of each stiffened gas law and *(iii)* Some mathematical constraints are needed for our model (convexity constraints…)

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