Joule Thomson Effect Inversion Curve
The differential coefficient ^ was first investigated by James Joule and William Thomson in the 1850s [23], before Thomson was elevated to the peerage, to become the first Lord Kelvin. So it is also referred to as the JouleKelvin coefficient.
It is a measure of the effect of the throttling process on a gas, when it is forced through a porous plug, or a small aperture or nozzle. The drop in pressure, at constant enthalpy H, has an effect on temperature. The enthalpy of hydrogen as a Van der Waals gas is given by the specific values (per mole) of the extensive variables H and V, written h and v.
f a ( RT a\ if v \ 2a h = u + pv = RT   + v   r = RWi +(4.24)
where u is the internal energy, f is the degree of freedom (5 for H2) and v is the molar volume of the gas. The change in enthalpy is given by the differential and is
Table 4.8 Coefficients C, as a function oftemperature
60 
3.54561 
io4 
1.66337 
IO7 
2.99498 
io 
l 
77 
1.38130) 
104 
4.67096 
<io8 
5.93690 
io 
2 
93.15 
3.86094 
IO5 
1.23153 
<io8 
9.00347 
io 
2 
113.15 
1.32755 
105 
1.01021 
<108 
4.43987 
io 
3 
133.15 
3.59307 
IO5 
5.40741 
<io9 
4.34407 
io 
3 
153.15 
4.24489 
IO5 
5.03665 
<io9 
8.93238 
io 
4 
173.15 
4.29174 
IO5 
5.56911) 
<io9 
2.11366 
io 
3 
193.15 
4.47329 
IO5 
3.91672 
<io9 
4.92797 
io 
4 
213.15 
4.34505 
IO5 
3.91417 
<io9 
1.50817 
io 
3 
233.15 
4.45773 
IO5 
2.18237 
<io9 
5.85180 
io 
4 
253.15 
4.48069 
IO5 
8.98684 
io10 
2.03650 
io 
3 
273.15 
4.25722 
IO5 
9.50702 
io10 
1.44169 
io 
3 
293.15 
3.69294 
IO5 
2.83279 
<io9 
1.93482 
io 
3 
298.15 
3.49641 
IO5 
3.60045 
<io9 
3.22724 ) 
io 
3 
313.15 
4.16186 
IO5 
5.28484 
IO10 
2.73571 
io 
3 
333.15 
4.05294 
IO5 
7.21562 
io10 
2.52962 
io 
3 
10 
15  
10 
15 
3.54211 ) 
< 10  
10 
15 
2.40671 ) 
119 Table 4.9 The volume V, enthalpy H, Gibbs free energy G and entropy S of nH2 at a temperature (T = 300 K) and pressures (p = 0.1 MPa100GPa, i.e. 1 x 106 bar)a
aThe values are taken from Hemme et al. [20]. The complete set of data for temperatures (T = 1002000 K) can be found in Fukai [21] and Sugimoto [22]. aThe values are taken from Hemme et al. [20]. The complete set of data for temperatures (T = 1002000 K) can be found in Fukai [21] and Sugimoto [22]. zero for an isoenthalpic expansion dK dh . In a pressuretemperature plot, for any gas, the locus of points at which the drop in pressure, at constant enthalpy H, has no effect on the temperature is called the inversion curve for that gas. So the inversion curve has the simple form ^ = 0. These curves are plotted by first finding a family of isoenthalpies (H = const.) on the Tp plane, then connecting their stationary points. Except along the inversion curve, throttling either heats the gas (¡x< 0), or cools it 0). Cooling of gases, in order to subsequently liquefy them, is usually accomplished by throttling in a region where it causes cooling (Figure 4.6). The specific values (per mole) of the extensive variables H, S and V, are written h, s and v. Then the specific enthalpy is [24] Thus it follows that the JouleThomson coefficient is dv ST T2 d CP ST The first thing to notice about the JouleThomson 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 JouleThomson coefficient (Equation (4.29)) leads for H = 0 and substituting index r by i to Substituting Vi with pi one finds (Figure 4.7) Ti = —2 (45  pi ± 12^9—^) The two branches join at p = 9, where T = 3 and V = 1. They connect this point with the T axis, where p vanishes, at the two points T =  and T = 27/4. And of course, inside this curve is the region in which the Van der Waals equation predicts that throttling any gas will produce cooling. 
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