A Joule-Thomson or Isenthalpic Expansion
A gas at pressure P_{1} \text { and temperature } T_{1} \text { is steadily exhausted to the atmosphere at pressure } P_{2} through a pressure-reducing valve. Find an expression relating the downstream gas temperature T_{2} \text { to } P_{1}, P_{2} \text {, and } T_{1}. Since the gas flows through the valve rapidly, one can assume that there is no heat transfer to the gas. Also, the potential and kinetic energy terms can be neglected.
The flow process is schematically shown in the figure. We will consider the region of space that includes the flow obstruction (indicated by the dashed line) to be the system, although, as in Illustration 3.2-1, a fixed mass of gas could have been chosen as well. The pressure of the gas exiting the reducing valve will be P2, the pressure of the surrounding atmosphere. (It is not completely obvious that these two pressures should be the same. However, in the laboratory we find that the velocity of the flowing fluid will always adjust in such a way that the fluid exit pressure and the pressure of the surroundings are equal.) Now recognizing that our system (the valve and its contents) is of constant volume, that the flow is steady, and that there are no heat or work flows and negligible kinetic and potential energy changes, the mass and energy balances (on a molar basis) yield
0=\dot{N}_{1}+\dot{N}_{2} \quad \text { or } \quad \dot{N}_{2}=-\dot{N}_{1}and
0=\dot{N}_{1} \underline{H}_{1}+\dot{N}_{2} \underline{H}_{2}=\dot{N}_{1}\left(\underline{H}_{1}-\underline{H}_{2}\right)Thus
\underline{H}_{1}=\underline{H}_{2}or, to be explicit,
\underline{H}\left(T_{1}, P_{1}\right)=\underline{H}\left(T_{2}, P_{2}\right) \quad \text { or } \quad \hat{H}\left(T_{1}, P_{1}\right)=\hat{H}\left(T_{2}, P_{2}\right)so that the initial and final states of the gas have the same enthalpy. Consequently, if we knew how the enthalpy of the gas depended on its temperature and pressure, we could use the known values of T_{1}, P_{1} \text {, and } P_{2} to determine the unknown downstream temperature T_{2}.
Comments
1. The equality of enthalpies in the upstream and downstream states is the only information we get from the thermodynamic balance equations. To proceed further we need constitutive information, that is, an equation of state or experimental data interrelating \underline{H}, T, and P. Equations of state are discussed in the following section and in much of Chapter 6.
2. The experiment discussed in this illustration was devised by William Thomson (later Lord Kelvin) and performed by J. P. Joule to study departures from ideal gas behavior. The Joule-Thomson expansion, as it is called, is used in the liquefaction of gases and in refrigeration processes (see Chapter 5).