Joule-Thomson inversion curve

Figure 3.7 (a) Joule-Thomson inversion curve (/o.t. = 0) for nitrogen, (b) The Joule -Thomson coefficient of nitrogen gas. At the lowest temperature, 123.15 K. nitrogen liquifies hence the curve for the gas terminates at the vapor pressure. 

It is shown at the end of Ex. 7.5 that the Joule/Thomson inversion curve is the locus of states for which (dZ/dT) p =0. We apply the following general equation of differential calculus ... 

FIGURE l.A.l Joule-Thomson inversion curve for nitrogen. 

The Joule-Thomson inversion curve as calculated from (1) is in some doubt. This is apparent at high temperatures where (1) predicts a maximum inversion temperature of 293 °K, whereas a more probable value would be 225 °K. Although the effect of such an uncertainty on the temperature-entropy diagram is expected to be small, further efforts are being made to reconcile this problem. 

THE JOULE-THOMSON INVERSION CURVE AND SONIC VELOCITIES... 

Figure A6 Joule-Thomson Inversion Curve for Carbon Dioxide as a Function of the Pressure and the Temperature.

The extrapolation behaviour of empirical multi-parameter equations of state has been summarized by Span and Wagner. " Aside from the representation of shock tube data for the Hugoniot curve at very high temperatures and pressures, an assessment of the extrapolation behaviour of an equation of state can also be based on the so called ideal curves that were first discussed by Brown. While reference equations of state generally result in reasonable estimates for the Boyle, ideal, and Joule-Thomson inversion curves, the prediction of reasonable Joule inversion curves is still a challenge. Equations may result in unreasonable estimates of Boyle, ideal and Joule-Thomson plots especially when the equations are based on limited experimental data. 

Joule-Thomson inversion curve for N2 gas actuai data as compared to predictions based on the van der Waals equation state. 

R.C. Hendricks et al.. Joule-Thomson inversion curves and related coefficients for several simple fluids, NASA Technical Note D-6807 (1972). 

Colina, C.M. Turrens, L.F. Olivera-Fuentes, C. Gubbins, K.E. Vega, L.F. (2002). Predictions of the Joule-Thomson Inversion Curve for the n-Alkane Series and Carbon Dioxide from the Soft-SAFT Equation of State. Ind. Eng. Chem. Res. 41,1069-1075. 

This observation is important because it also permits us to conclude that it will not be possible to construct a tangent through the origin at any point of the curve B2 T) no matter whether we consider a bulk or confined ideal quantum gas and irrespective of whether the quantum particles are Fermions or Bosons. In other words, for the ideal (bulk and confined) quantum gases, an inversion temperature Tj v does not exist because Eq. (5.155) does not have a solution. However, the reader should note that a Joule-Thomson effect does exist as pointed otit in Section 5.7.1, namely a dilute gas of Bosons is always cooled upon an iscnthalpie expansion B2 T) < 0), whereas a gas of Fermions is always heated during this process B2 (T) > 0). Tire extent to which this happens is modified in a nontrivial way by confinement according to the above discussion. 

The first thing to notice about the Joule-Thomson coefficient is that it vanishes for an ideal gas. The inversion curve is everywhere for an ideal gas For real gases, the isoenthalpies have to be determined experimentally and the line connecting the stationary points is the inversion curve. 

The simplest equation of state for a gas which predicts an inversion curve is the Van der Waals equation. It yields an inversion curve, that is fairly close, but not exactly correct. The Van der Waals equation in reduced variables (Equation (4.14)) used in the equation for the Joule-Thomson coefficient (Equation (4.29)) leads for IX = 0 and substituting index r by i to... 

This means that there is a curve that represents the points where Joule-Thomson coefficient is exactly zero and it called the JT inversion curve. This curve is shown in Figure A6. At temperatures and pressure below this curve the Joule-Thomson coefficient is positive and above it is negative. 

If one were to compare this Figure A6 to Figure A3 one would see that the inversion curve would be only in a small region in the left-hand top corner. This means the Joule-Thomson coefficient is negative only for high density carbon dioxide. 

For any gas, the sign of the Joule-Thomson effect depends on tanperature and pressure. The positive effect for each gas is observed only in the limited interval of temperatures and pressures. For each gas there are values of temperature and pressure at which the Joule-Thomson effect is equal to zero (no temperature changes occur at gas expansion in vacuum). These points (T, p,) are called points of inversion. At these points, the influence of forces of attraction is completely compensated for by the influence of repulsion forces consequently the gas temperature does not change. The set of inversion points forms an inversion curve in a p-T diagram. 

Figure 3.27 presents the inversion curve for nitrogen. It can be seen that, to a given value p, two points of inversion can occur. The curve of inversion outlines two points of inversion for which a positive Joule-Thomson effect is observed. Values for the upper and lower inversion points for some gases at various pressures are given in Table 3.2. For the majority of gases, the upper point of inversion lies above room temperature. Hydrogen and helium are an exception. 

Figure 3,27 An inversion curve of the Joule-Thomson effect. Table 3,2...

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