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SAFPWR mathematical model description

The previous pages were intended, before digging into the details, to get the reader familiar, by means of some simple examples, with the general features of the program, the language conventions, the input and output interfaces.
Let us now systematically present the various constitutive parts of the program.

Properties of water and steam.

Properties of sub-cooled water

Correlation linking specific volume, enthalpy and pressure
The solution method for the m,h,b-balances equations requires representing the correlation as an implicit, derivable function
f(h,vm,p)=0 (1)
where vm (2) is the specific (unit-mass) volume.
The coefficients c0:c4 are obtained by least-square fitting on VDI tables over a range of (vm, h, p) which should bracket the range of values encountered during the analyzed transients.
vm(h,p) is calculated as the positive root of (1) solved for vm.
h(vm,p), as the root of (1) solved for h.
The partial derivatives
vmhp ≡(∂vm/∂h)p and
vmph ≡(∂vm/∂p)h can easily be calculated by total differentiating (1).
The isentropic pressure total derivative vmps along an adiabatic transformation is obtained as follows:
(3) is pressure total derivative along any transformation, but (4)holds for an isentropic transformation;
combining (3) and (4) provides vmps (5).
Chart 1 (pwr\t&h\t&h.123) compares the fitted values vm(h,p) with VDI reference and gives (on the second Y-axis) the relative error for a set of isobars from 30 to 180 bars. On each isobar, the rightmost point corresponds to saturated liquid. The relative error remains well within a 1% range.
Water Temperature
The function t(h,p) is represented by the Kaisermann correlation /4 (6), which is quite accurate.
Coefficients atp, btp,... are linear functions (7:9) of p, whose coefficients may be found in (/4).
By differentiating (6), we also get thp (10) and its inverse, the specific heat htp (11).
Chart 2 and chart 3 and compare the correlated t(h,p) with the VDI reference for the same range of pressure and enthalpy. Right most points correspond to saturated water.
Max error is 1.5 C

Properties of over-heated steam

We use for vm(h,p) correlations of the same form ((1):15) as for sub cooled water.
Chart 4 (pwr\t&h\eausursa.123) compares v(h,p) correlation to VDI.
The relative error remains also within the 1 % range.
On each isobar the left-most point is on the saturated vapor line.
Chart 5 compares to t(VDI) to t(h,p).
Left-most points corresponds to saturated steam.
The error range (-1.5e-2, +3e-3) is slightly larger than for the sub cooled data.
Properties of saturated water-vapor mixture

Garland correlations/5 have been selected because they are very accurate and easily derivable.

They provide, along the saturated liquid line "l", the unit-mass volume vmln (noted here vln to make short; n=0:l9 is the subsystem index: 0 for primary, n for SG n)), the enthalpy hln, as well as, by derivation, their pressure derivatives
vlpn ≡ d vln(p) / d p and
hlpn ≡ d hln(p)/d p.
Similarly, along the saturated vapor line "v" we get
vvn(p), hvn(p) and pressure derivatives vvpn and hvpn.
To obtain the properties for vapor-water mixture of enthalpy h, we start from the general relation (16) with hlvn defined by (17).
By total differentiating (18) of (16) it is easy to calculate the partial derivatives:(19,20), where vpon, vphn and vpvn are pressure-only dependent variables defined by (21:23)
All the properties with suffix n depend only of pressure pn in subsystem n and are updated whenever pn is updated.
For a vapor-water mixture of enthalpy h and pressure pn we get (24), or simpler, in terms of mixture quality (24)
x ≡ (h - hln) / hlvn.
From the generic relation (5) the iso-entropy pressure derivative is (28).
For a mixture of mass m and volume v, multiplying (23) by m, the iso-s pressure derivative is (29):

Resolving discontinuities when crossing the saturation lines.

The jump observed between the accurate Garlant value vl(p) and the less accurate sub-cooled correlation value vm[hl(p), p] must be offset to avoid numerical problems.

To force continuity, we set (30).
The resulting error (31) can be edited for information.
Likewise we will assume (32) as the saturated liquid temperature where t[hl(p), p] is the Kaisermann approximation (6).
The same assumptions are retained for vv(p) and tv(p).
The effect of the resulting error on vl(p) is deemed acceptable considering the uncertainty on components water mass and volume.
The difference tv(p) - tl(p) is of course non-physical.
In order to accommodate for it, we will assume that t(h,p) follows the same x-dependency (33) as vm(h,p).