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 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

2.42574 : -3.24527 : -2.63262 :

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

p [bar] Vm [cm3 mol-1] H [)mol-1] G [)mol-1] S [)mol-1K-1]

1

24943.02

8506

-30724

130.77

2

12485.87

8507

-28994

125

5

5003.08

8510

-26704

117.38

10

2508.87

8515

-24968

111.61

20

1261.83

8526

-23224

105.84

50

513.73

8560

-20895

98.18

100

264.51

8620

-19091

92.37

200

140.09

8747

-17210

86.52

500

65.78

9176

-14454

78.77

1000

40.98

9954

-11924

72.93

2000

27.96

11529

-8615

67.15

5000

18.75

15896

-1962

59.53

10000

14.58

22319

6189

53.77

20000

11.56

33414

19013

48

50000

8.64

60517

48402

40.39

100000

6.84

94585

86094

28.31

200000

5.52

154070

146981

23.63

500000

4.1

293183

287378

19.35

1000000

3.23

472676

467424

17.51

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]

Joule Thomson Koeffizient Co2
Fig. 4.6 The Joule-Thomson coefficient for H2 in the Van der Waals approximation (Equation (4.26)) at a pressure of p = 0.1 MPa (solid line) and p = 10 MPa (dashed line).

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—^)

Inversion Curve For Air
Fig. 4.7 Inversion curve of n-hydrogen and calculated inversion curve (Equation (4.31)) using the critical pressure pk = 1.325 MPa and the critical temperature Tk = 33.19 K for hydrogen.

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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Responses

  • BELL
    How to produce inversion curve from joule thomson?
    7 months ago

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