Temperature Joule-Thomson coefficients

Figure 3.7(a) shows experimental values of fj,j T obtained for N gas, while Figure 3.7(b) shows how the Joule-Thomson coefficient for N2 gas changes with pressure and temperature.2... 

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. 

The Joule-Thomson coefficient p.jx, is positive when a cooling of the gas (a temperature drop) is observed because dP is always negative, p.j x, will be positive when dT is negative. Conversely, p.j x, is a negative quantity when the gas warms on expansion because dT then is a positive quantity. Values of the Joule-Thomson coefficient for argon and nitrogen at several pressures and temperatures are listed in Table 5.5. 

It frequently is necessary to express the Joule-Thomson coefficient in terms of other partial derivatives. Considering the enthalpy as a function of temperature and pressure H T, P), we can write the total differential... 

Joule-Thomson Inversion Temperature. The Joule-Thomson coefficient is a function of temperature and pressure. Figure 5.8 shows the locus of points on a temperature-pressure diagram for which p,jx. is zero. Those points are at the Joule-Thomson inversion temperature 7). It is only inside the envelope of this... 

Because of this relationship between (TT — and p-j x.. the former quantity frequently is referred to as the Joule-Thomson enthalpy. The pressure coefficient of this Joule-Thomson enthalpy change can be calculated from the known values of the Joule-Thomson coefficient and the heat capacity of the gas. Similarly, as (H — is a derived function of the fugacity, knowledge of the temperature dependence of the latter can be used to calculate the Joule-Thomson coefficient. As the fugacity and the Joule-Thomson coefficient are both measures of the deviation of a gas from ideahty, it is not surprising that they are related. 

Although the van der Waals equation is not the best of the semi-empirical equations for predicting quantitatively the PVT behavior of real gases, it does provide excellent qualitative predictions. We have pointed out that the temperature coefficient of the fugacity function is related to the Joule-Thomson coefficient p,j x.- Let us now use the van der Waals equation to calculate p,j.T. from a fugacity equation. We will restrict our discussion to relatively low pressures. 

As Cpm is positive, the sign of the Joule-Thomson coefficient depends on the sign of the expression in parentheses in Equations (10.79) and (10.80). The expression in Equation (10.79) is a quadratic in T, and are two values of T exist at any value of P for which p.j x, = 0. Thus, Equation (10.79) predicts two values of the Joule-Thomson inversion temperature T,- for any pressure low enough for Equation (10.75) to be a good approximation for a. As we saw in Section (5.2) and Figure 5.8, this prediction fits, at least qualitatively, the experimental data for the Joule-Thomson experiment for N2 at low pressure. 

Joule Thomson coefficient m may be defined as the temperature change in degrees produced by a drop of one atmospheric pressure when the gas expands under conditions of constant enthalpy. It is expressed as... 

The measured Joule-Thomson coefficient /jljt provides valuable information about how the enthalpy of real gases depends on variables other than temperature. To obtain information about the P dependence of H, we can employ the Jacobi cyclic identity (1.14b) to rewrite the Joule-Thomson coefficient as... 

Figure 3.12 Qualitative temperature and pressure dependence of the Joule-Thomson coefficient MotO P) for C02.

The Joule-Thomson coefficient is the slope of the isenthalpic lines in the P-T projection. In the region where iJt<0, expansion through the valve (a decrease in pressure) results in an increase in temperature, whereas in the region where pJt >0, expansion results in a reduction in temperature. The latter area is recommendable for applying the PGSS process. 

We conclude that the Joule-Thomson coefficient is a function of both the temperature and the pressure, but, unlike the Joule coefficient, it does not go to zero as the pressure goes to zero. The inversion temperature, the temperature at which fi,T = 0, is also a function of the pressure. The value usually reported in the literature is the limiting value as the pressure goes to zero. 

There are two variations of the basic set-up of the Joule-Thomson experiment which both yield practical information. In the isothermal Joule-Thomson experiment the temperature is held constant with a downstream heater, and the resultant heat input for the pressure decrease permits an experimental evaluation of (8H/8P)T, the isothermal Joule-Thomson coefficient. In the other variation there is no throttling device used, and the pressure is held constant. For the steady-state flow of gas the temperature change is measured for measurable inputs of heat. This experiment, of course, yields (8H/8T)P, or CP. Thus, the variations of this constant-flow experiment can yield all three of the important terms in Equation (7.46). 

The Joule-Thomson coefficient is the slope of the isoenthalp and is a function of both temperature and pressure. From Eq. (23) and Eq. (7) of Appendix A, we can write... 

Heat capacity, molar Heat capacity at constant pressure Heat capacity at constant volume Helmholtz energy Internal energy Isothermal compressibility Joule-Thomson coefficient Pressure, osmotic Pressure coefficient Specific heat capacity Surface tension Temperature Celsius... 

The Joule Thomson coefficient is the ratio of the temperature decrease to the pressure drop, and is expressed in terms of the thermal expansion coefficient and the heat capacity... 

For a van der Waals gas determine the Joule Thomson coefficient and inversion temperature Ti in terms of a, b, P, T, and Cp. Make a sketch of versus T note and comment on the double-valued nature of the function. 

The quantity /x defined by this equation is known as the Joule-Thomson coefficient. It represents the limiting value of the experimental ratio of temperature difference to pressure difference as the pressure difference approaches zero ... 

The denominator on the right side of Eq. (4) is the heat capacity at constant pressure Cp. The numerator is zero for an ideal gas [see Eq. (1)]. Accordingly, for an ideal gas the Joule-Thomson coefficient is zero, and there should be no temperature difference across the porous plug. Eor a real gas, the Joule-Thomson coefficient is a measure of the quantity [which can be related thermodynamically to the quantity involved in the Joule experiment, Using the general thermodynamic relation ... 

For most gases under ordinary conditions, 2a RT > b (the attractive forces predominate over the repulsive forces in determining the nonideal behavior) and the Joule-Thomson coefficient is therefore positive (gas cools on expansion). At a sufficiently high temperature, the inequality is reversed, and the gas warms on expansion. The temperature at which the Joule-Thomson coefficient changes sign is called the inversion temperature Tj. For a van der Waals gas,... 

The Joule-Thomson coefficient p,j-r describes the extent and direction of the temperature change for an isenthalpic change of state (constant enthalpy h) ... 

The liquefaction of helium by a controlled expansion process necessitates preliminary cooling because its Joule-Thomson coefficient is negative (spontaneous expansion heats the gas) down to an inversion temperature of 40 All the gases have C /C ratios very close to 5/3, the theoretical value for an ideal monatomic gas. The elements are liquid over very small temperature ranges. Plelium can be solidified only under pressure under 26 atmospheres it solidifies at 0.9 °K. 

The quantity dT/dP)u is called the Joule-Thomson coefficient, and is represented by the symbol mj.t. it is equal to the rate of change of temperature with the pressure in a streaming process through a plug or throttle. According to equation (9.11), dH/d2 )p is the heat capacity of the gas at constant pressure, i.e., Cp, so that (11.4) is equivalent to... 

Equation (11.6) is quite general and should apply to any gas/ for its derivation is based entirely on the first la>y of thermodynamics without assuming any specific properties of the system. However, for an ideal gas, (dE/dV)r is zero, as seen earlier, and since PV = /2T, it follows that td PV)/dP ]T is also zero hence, since Cp is finite, it is seen from equation (11.6) that for an ideal gas mj.t. must be zero.f The Joule-Thomson coefficient of an ideal gas should thus be zero, so that there should be no change of temperature when such a gas e.xpands through a throttle. J... 

---