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MATHEMATICAL MODELS. Valve modelling
The model valve is intended for calculation of a He mass flow through cryogenic elements, such as valves, holes, gaps, etc. It is assumed that mass flow through the valve is forced by pressure difference in pare collectors to which it is connected.
To associate the mass flow through the valve with the thermodynamic properties of helium in both collectors we use a simplest classical conception. The flow through the valve is modelled as isentropic expansion (dH-dP/r = 0) of the compressible fluid from the inlet pressure (before the valve) to the outlet pressure. Depending on a pressure drop between the inlet/outlet the outlet valve pressure is equal to the outside pressure (sub-critical flow out) or critical pressure (critical flow out, the outlet velocity is equal to the local sonic speed). In both cases, the full enthalpy (h+u2/2) of the flow under the isentropic expansion is conservative, that allows calculation of all thermodynamic characteristics of the outlet flow. Two basic parameters are assigned for the valve: the minimum
cross-section area A of the valve (which depends on the valve lift) and correction factor m to take into account a non-isentropic expansion.
For a sub-critical flow the pressure at the valve outlet is equal to the pressure in the outlet collector. In the case of the critical flow the outlet pressure is equal to the critical one. So, for calculation of helium properties in the valve outlet k the following system of equations is used:
where Hk, Uk, Pk, rk is the enthalpy, velocity, pressure and density of helium at the valve outlet; HWin, SWin is the enthalpy and entropy of helium in the inlet collector; PWout is the pressure in the outlet collector and Pkcrit
is the critical pressure in the valve outlet. For a real non-isentropic flow the correction factor mk < 1 is
used.
The set of additional parameters allows control of the valve lift depending on the pressure variations in adjoining collectors applicable for modelling the multi-purpose valves.
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