Ethylene from a pipeline (50 t/h, 70 bar, 5°C) passes a heat exchanger and is heated up to 45°C. The pressure drop is negligible. As usual, in the data sheet of the heat exchanger the cP values at inlet and outlet are given:
cP ( 5°C, 70 bar) = 3.984 J/g*K
cP (45°C, 70 bar) = 3.783 J/g*K
Calculate the heat to be exchanged
a) by estimation with an average cP
b) using a high-precision equation of state
Which solution shall be preferred?
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:
To calculate aid, an expression for the ideal gas heat capacity is required. In Appendix A parameters for the following expressions are given:
The resulting expressions for aid can be derived using eq. 2.105 and the antiderivatives given in Appendix C (F4 - Antiderivatives of cPid correlations):
The constants cancel out when using the expressions in eq. 2.105, which also adds the value at a reference point. Usually, href and sref are set to be zero at the ideal gas state at Tref = 298.15 K and Pref = 101325 Pa, even if this reference point is fictitious and the fluid regarded is in the liquid state.
The residual part is different for non-polar and polar fluids:
For non-polar fluids (methane, ethane, propane, n-butane, n-pentane, n-hexane, n-heptane, n-octane, argon, oxygen, nitrogen, ethylene, isobutane, cyclohexane, sulfur hexafluoride, carbon monoxide, carbonyl sulfide, n-decane, hydrogen sulfide, isopentane, neopentane, isohexane, krypton, n-nonane, toluene, xenon and R116):
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):
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).
Component Selection and Model Parameters:
Parameter vectors for the ideal gas heat capacity (cid) and for the real part of the Helmholtz energy (c) are stored in separate files and are imported as references. In the ideal heat capacity parameter vector, the first element denotes the equation to be used (1 - 3.70, 2 - 3.69).
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.
a) In the first step the mean heat capacity in the temperature range between 5 and 45°C is
Although in the supercritical state, the root function does not succeed to find 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.
In the next step the molar and total enthalphy difference between inlet and outlet stream is calculated:
There is a large difference between the two results. Although the cP values are correct, approach a) does not yield a reasonable result. The problem is that cP has a sharp maximum in the area in the vicinity of the critical point, as the following plot illustrates: