Theory
Like reference equations, state-of-the-art technical equations of state are formulated in terms of the Helmholtz energy, which is split into an ideal gas part and a residual part. Instead of the specific volume, the density is used as a variable:
2.104
with
To calculate
3.69
The resulting expressions for
2.105
For eq. 3.70 the following antiderivatives are given:
C.197
C.198
For eq. 3.69 the antiderivatives are:
C.195
C.196
The residual part is different
For polar fluids (R11, R12, R22, R32, R113, R123, R125, R134a, R143a, R152a, carbon dioxide, ammonia, acetone, nitrous oxide, sulfur dioxide, R141b, R142b, R218 and R245fa):
References:
R. Span; W. Wagner; Int. J. Thermophys. 24(1), 41-109 (2003).
E. W. Lemmon; R. Span; J. Chem. Eng. Data 24(1), 41-109 (2006).
R. Span; W. Wagner; Int. J. Thermophys. 24(1), 111-162 (2003).
E. C. Ihmels; E. W. Lemmon; Fluid Phase Equilib. 207, 111-130 (2003).
cP
id function 3.70 and the two different integrals required for calculation of
cP
id function 3.69 and the two different integrals required for calculation of
Set the correct functions for cP id and the two different integrals depending on the function used.
Auxiliary functions:
pressure
A starting value close above the hardcore volume is usually possible and ensures finding the liquid root. As P(v) covers several orders of magnitude, the objective of the root function should be the relative deviation in pressure.
Solution:
Conditions:
In the first step the molar volume of ethylene at the conditions inside the pressurized bottle has to be calculated:
(complex solution)
Although in the supercritical state, the root function found only a complex and not a real solution when starting from the ideal gas volume. The second function usually used to find the liquid volume starts at a very low liquid volume and converges to the correct solution.
Dividing the volume of the bottle by the molar volume yields the number of moles, that can be easily converted to the mass of ethylene in the bottle:
The pressurized bottle contains 20.523 kg of ethylene.
The following plot shows the dependence of the objective function of the root function on reduced volume using the high precision equation of state for ethylene: