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 Joule-Kelvin 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 |
io-4 |
1.66337 |
IO-7 |
-2.99498 |
io- |
l |
77 |
-1.38130) |
10-4 |
4.67096 |
<io-8 |
5.93690 |
io- |
2 |
93.15 |
-3.86094 |
IO-5 |
1.23153 |
<io-8 |
9.00347 |
io- |
2 |
113.15 |
1.32755 |
10-5 |
1.01021 |
<10-8 |
4.43987 |
io- |
3 |
133.15 |
3.59307 |
IO-5 |
5.40741 |
<io-9 |
4.34407 |
io- |
3 |
153.15 |
4.24489 |
IO-5 |
5.03665 |
<io-9 |
8.93238 |
io- |
4 |
173.15 |
4.29174 |
IO-5 |
5.56911) |
<io-9 |
-2.11366 |
io- |
3 |
193.15 |
4.47329 |
IO-5 |
3.91672 |
<io-9 |
-4.92797 |
io- |
4 |
213.15 |
4.34505 |
IO-5 |
3.91417 |
<io-9 |
-1.50817 |
io- |
3 |
233.15 |
4.45773 |
IO-5 |
2.18237 |
<io-9 |
5.85180 |
io- |
4 |
253.15 |
4.48069 |
IO-5 |
8.98684 |
io-10 |
2.03650 |
io- |
3 |
273.15 |
4.25722 |
IO-5 |
9.50702 |
io-10 |
1.44169 |
io- |
3 |
293.15 |
3.69294 |
IO-5 |
2.83279 |
<io-9 |
-1.93482 |
io- |
3 |
298.15 |
3.49641 |
IO-5 |
3.60045 |
<io-9 |
-3.22724 ) |
io- |
3 |
313.15 |
4.16186 |
IO-5 |
-5.28484 |
IO-10 |
2.73571 |
io- |
3 |
333.15 |
4.05294 |
IO-5 |
-7.21562 |
io-10 |
2.52962 |
io- |
3 |
10- |
-15 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
10- |
-15 |
3.54211 ) |
< 10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
10- |
-15 |
2.40671 ) |
1-19 Table 4.9 The volume V, enthalpy H, Gibbs free energy G and entropy S of n-H2 at a temperature (T = 300 K) and pressures (p = 0.1 MPa-100GPa, i.e. 1 x 106 bar)a
aThe values are taken from Hemme et al. [20]. The complete set of data for temperatures (T = 100-2000 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 = 100-2000 K) can be found in Fukai [21] and Sugimoto [22]. zero for an isoenthalpic expansion dK dh . In a pressure-temperature 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 T-p 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 Joule-Thomson coefficient is dv ST T2 d CP ST 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 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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