Expansion, isothermal

Condensable Hquids also are recovered from high pressure gas reservoirs by retrograde condensation. In this process, the high pressure fluid from the reservoir produces a Hquid phase on isothermal expansion. As the pressure decreases isotherm ally the quantity of the Hquid phase increases to a maximum and then decreases to disappearance. In the production of natural gas Hquids from these high pressure wells, the well fluids are expanded to produce the optimum amount of Hquid. The Hquid phase then is separated from the gas for further processing. The gas phase is used as a raw material for one of the other recovery processes, as fuel, or is recompressed and returned to the formation. 

Other Refrigeration Methods. Cryocoolers provide low temperature refrigeration on a smaller scale by a variety of thermodynamic cycles. The Stirling cycle foUows a path of isothermal compression, heat transfer to a regenerator matrix at constant volume, isothermal expansion with heat transfer from the external load at the refrigerator temperature, and finally heat transfer to the fluid from the regenerator at constant volume. 

Isothermal change A process that takes place at constant temperature, such as the isothermal expansion of a gas. 

Figure 2-31. P-V process diagrams (a) isothermal expansion (b) isobaric compression.

Illustrative Example.—Consider the isothermal expansion of an ideal gas between fixed limits of volume ... 

The isothermal expansion of an ideal gas is an aschistic process.— If a mass of gas expands isothermally, the heat absorbed is equal to the external work done. 

Figure 2.5 Comparison of the work accomplished following four different paths during an isothermal expansion.

Figure 2.6 Comparison of the work obtained from the two isothermal expansions at 300 K of ideal gas following path (i) in (a) and path (ii) in (b). In each instance, the initial pressure is 0.200 MPa and the final pressure is 0.100 MPa.

Example 2.1 Compare the work obtained from the two isothermal expansions at 300 K of one mole of ideal gas following the paths shown in Figure 2.6. 

Solution AC = 0 for the isothermal expansion of the ideal gas. Hence, from equation (2.33)... 

In summary, in the isothermal expansion of ideal gas, work flowing out of a system is balanced by heat flowing into the system so that AC = 0. 

Calculation of AS for the Reversible Isothermal Expansion of an Ideal Gas Integration of equation (2.38) gives... 

From example 2.3 we saw that for the reversible isothermal expansion of ideal gas... 

This example provides the proof that we promised earlier that (dU/dV)T = 0 for the ideal gas. We also obtain the same equation for AS as we derived earlier using AS = qrev/T. It also shows that AH — 0 for the isothermal expansion of ideal gas and gives a thermodynamically consistent valuef for AG. That is, in the final state... 

It is useful to compare the reversible adiabatic and reversible isothermal expansions of the ideal gas. For an isothermal process, the ideal gas equation can be written... 

Figure 3.2 compares a series of reversible isothermal expansions for the ideal gas starting at different initial conditions. Note that the isotherms are parallel. They cannot intersect since this would give the gas the same pressure and volume at two different temperatures. Figure 3.3 shows a similar comparison for a series of reversible adiabatic expansions. Like the isotherms, the adiabats cannot intersect. To do so would violate the Caratheodory principle and the Second Law of Thermodynamics, since the gas would have two different entropies at the same temperature, pressure, and volume. 

A reversible isothermal expansion of the ideal gas is made from an initial volume V to a volume Vz at an absolute (ideal gas) temperature 73. The amount of pressure-volume work in done by the system is obtained by substituting into Equation (2.16). The result is... 

To maintain isothermal conditions during this process, a quantity of heat qi is absorbed from a high-temperature heat reservoir operating at T2. Since AC/ = 0 for this isothermal expansion, qz — -in, so that... 

For a horizontal pipe, dz = 0, and for isothermal expansion of an ideal gas dH = 0. Thus if the system does no work on the surroundings ... 

In a synthetic ammonia plant the hydrogen is fed through a 50 mm steel pipe to the converters. The pressure drop over the 30 m length of pipe is 500 kN/m2, the pressure at the downstream end being 7.5 MN/m2. What power is required in order to overcome friction losses in the pipe Assume isothermal expansion of the gas at 298 K. What error is introduced by assuming the gas to be an incompressible fluid of density equal to that at the mean pressure in the pipe /r — 0.02 mN s/m2. 

To calculate the work of reversible, isothermal expansion of a gas, we have to use calculus, starting at Eq. 3 written for an infinitesimal change in volume, dV ... 

EXAMPLE 6.2 Sample exercise Calculating the work of isothermal expansion... 

SOLUTION For expansion against constant external pressure we use Eq. 3 and for reversible, isothermal expansion we use Eq. 4 ... 

The work done by any system on its surroundings during expansion against a constant pressure is calculated from Eq. 3 for a reversible, isothermal expansion of an ideal gas, the work is calculated from Eq. 4. A reversible process is a process that can be reversed by an infinitesimal change in a variable. 

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